Difference between revisions of "031 Review Part 1"
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<div class="noautonum">__TOC__</div> | <div class="noautonum">__TOC__</div> | ||
| − | == [[ | + | == [[031_Review Part 1,_Problem_1|<span class="biglink"><span style="font-size:80%"> Problem 1 </span></span>]] == |
<span class="exam">True or false: If all the entries of a <math style="vertical-align: 0px">7\times 7</math> matrix <math style="vertical-align: 0px">A</math> are <math style="vertical-align: -4px">7,</math> then <math style="vertical-align: 0px">\text{det }A</math> must be <math style="vertical-align: 0px">7^7.</math> | <span class="exam">True or false: If all the entries of a <math style="vertical-align: 0px">7\times 7</math> matrix <math style="vertical-align: 0px">A</math> are <math style="vertical-align: -4px">7,</math> then <math style="vertical-align: 0px">\text{det }A</math> must be <math style="vertical-align: 0px">7^7.</math> | ||
| − | == [[ | + | == [[031_Review Part 1,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] == |
<span class="exam"> True or false: If a matrix <math style="vertical-align: 0px">A^2</math> is diagonalizable, then the matrix <math style="vertical-align: 0px">A</math> must be diagonalizable as well. | <span class="exam"> True or false: If a matrix <math style="vertical-align: 0px">A^2</math> is diagonalizable, then the matrix <math style="vertical-align: 0px">A</math> must be diagonalizable as well. | ||
| − | == [[ | + | == [[031_Review Part 1,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] == |
<span class="exam">True or false: If <math style="vertical-align: 0px">A</math> is a <math style="vertical-align: -1px">4\times 4</math> matrix with characteristic equation <math style="vertical-align: -5px">\lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,</math> then <math style="vertical-align: 0px">A</math> is diagonalizable. | <span class="exam">True or false: If <math style="vertical-align: 0px">A</math> is a <math style="vertical-align: -1px">4\times 4</math> matrix with characteristic equation <math style="vertical-align: -5px">\lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,</math> then <math style="vertical-align: 0px">A</math> is diagonalizable. | ||
| − | == [[ | + | == [[031_Review Part 1,_Problem_4|<span class="biglink"><span style="font-size:80%"> Problem 4 </span>]] == |
<span class="exam"> True or false: If <math style="vertical-align: 0px">A</math> is invertible, then <math style="vertical-align: 0px">A</math> is diagonalizable. | <span class="exam"> True or false: If <math style="vertical-align: 0px">A</math> is invertible, then <math style="vertical-align: 0px">A</math> is diagonalizable. | ||
| − | == [[ | + | == [[031_Review Part 1,_Problem_5|<span class="biglink"><span style="font-size:80%"> Problem 5 </span>]] == |
<span class="exam">True or false: If <math style="vertical-align: 0px">A</math> and <math style="vertical-align: 0px">B</math> are invertible <math style="vertical-align: 0px">n\times n</math> matrices, then so is <math style="vertical-align: -1px">A+B.</math> | <span class="exam">True or false: If <math style="vertical-align: 0px">A</math> and <math style="vertical-align: 0px">B</math> are invertible <math style="vertical-align: 0px">n\times n</math> matrices, then so is <math style="vertical-align: -1px">A+B.</math> | ||
| − | == [[ | + | == [[031_Review Part 1,_Problem_6|<span class="biglink"><span style="font-size:80%"> Problem 6 </span>]] == |
<span class="exam"> True or false: If <math style="vertical-align: 0px">A</math> is a <math style="vertical-align: 0px">3\times 5</math> matrix and <math style="vertical-align: -4px">\text{dim Nul }A=2,</math> then <math style="vertical-align: 0px">A\vec{x}=\vec{b}</math> is consistent for all <math style="vertical-align: 0px">\vec{b}</math> in <math style="vertical-align: 0px">\mathbb{R}^3.</math> | <span class="exam"> True or false: If <math style="vertical-align: 0px">A</math> is a <math style="vertical-align: 0px">3\times 5</math> matrix and <math style="vertical-align: -4px">\text{dim Nul }A=2,</math> then <math style="vertical-align: 0px">A\vec{x}=\vec{b}</math> is consistent for all <math style="vertical-align: 0px">\vec{b}</math> in <math style="vertical-align: 0px">\mathbb{R}^3.</math> | ||
| − | == [[ | + | == [[031_Review Part 1,_Problem_7|<span class="biglink"><span style="font-size:80%"> Problem 7 </span>]] == |
<span class="exam">True or false: Let <math style="vertical-align: 0px">C=AB</math> for <math style="vertical-align: 0px">4\times 4</math> matrices <math style="vertical-align: 0px">A</math> and <math style="vertical-align: 0px">B.</math> If <math style="vertical-align: 0px">C</math> is invertible, then <math style="vertical-align: 0px">A</math> is invertible. | <span class="exam">True or false: Let <math style="vertical-align: 0px">C=AB</math> for <math style="vertical-align: 0px">4\times 4</math> matrices <math style="vertical-align: 0px">A</math> and <math style="vertical-align: 0px">B.</math> If <math style="vertical-align: 0px">C</math> is invertible, then <math style="vertical-align: 0px">A</math> is invertible. | ||
| − | == [[ | + | == [[031_Review Part 1,_Problem_8|<span class="biglink"><span style="font-size:80%"> Problem 8 </span>]] == |
<span class="exam">True or false: Let <math style="vertical-align: 0px">W</math> be a subspace of <math style="vertical-align: 0px">\mathbb{R}^4</math> and <math style="vertical-align: 0px">\vec{v}</math> be a vector in <math style="vertical-align: 0px">\mathbb{R}^4.</math> If <math style="vertical-align: 0px">\vec{v}\in W</math> and <math style="vertical-align: -4px">\vec{v}\in W^\perp,</math> then <math style="vertical-align: 0px">\vec{v}=\vec{0}.</math> | <span class="exam">True or false: Let <math style="vertical-align: 0px">W</math> be a subspace of <math style="vertical-align: 0px">\mathbb{R}^4</math> and <math style="vertical-align: 0px">\vec{v}</math> be a vector in <math style="vertical-align: 0px">\mathbb{R}^4.</math> If <math style="vertical-align: 0px">\vec{v}\in W</math> and <math style="vertical-align: -4px">\vec{v}\in W^\perp,</math> then <math style="vertical-align: 0px">\vec{v}=\vec{0}.</math> | ||
| − | == [[ | + | == [[031_Review Part 1,_Problem_9|<span class="biglink"><span style="font-size:80%"> Problem 9 </span>]] == |
<span class="exam">True or false: If <math style="vertical-align: 0px">A</math> is an invertible <math style="vertical-align: 0px">3\times 3</math> matrix, and <math style="vertical-align: 0px">B</math> and <math style="vertical-align: 0px">C</math> are <math style="vertical-align: 0px">3\times 3</math> matrices such that <math style="vertical-align: -4px">AB=AC,</math> then <math style="vertical-align: 0px">B=C.</math> | <span class="exam">True or false: If <math style="vertical-align: 0px">A</math> is an invertible <math style="vertical-align: 0px">3\times 3</math> matrix, and <math style="vertical-align: 0px">B</math> and <math style="vertical-align: 0px">C</math> are <math style="vertical-align: 0px">3\times 3</math> matrices such that <math style="vertical-align: -4px">AB=AC,</math> then <math style="vertical-align: 0px">B=C.</math> | ||
Revision as of 12:06, 9 October 2017
This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.
Click on the boxed problem numbers to go to a solution.
Problem 1
True or false: If all the entries of a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7\times 7} matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7,} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }A} must be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7^7.}
Problem 2
True or false: If a matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^2} is diagonalizable, then the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} must be diagonalizable as well.
Problem 3
True or false: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\times 4} matrix with characteristic equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is diagonalizable.
Problem 4
True or false: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is invertible, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is diagonalizable.
Problem 5
True or false: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} are invertible Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times n} matrices, then so is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A+B.}
Problem 6
True or false: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 5} matrix and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{dim Nul }A=2,} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\vec{x}=\vec{b}} is consistent for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{b}} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3.}
Problem 7
True or false: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C=AB} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\times 4} matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B.} If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is invertible, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is invertible.
Problem 8
True or false: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} be a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^4} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}} be a vector in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^4.} If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}\in W} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}\in W^\perp,} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}=\vec{0}.}
Problem 9
True or false: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is an invertible Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3} matrix, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3} matrices such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB=AC,} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B=C.}