031 Review Part 2

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 

Consider the matrix  $A={\begin{bmatrix}1&-4&9&-7\\-1&2&-4&1\\5&-6&10&7\end{bmatrix}}$  and assume that it is row equivalent to the matrix

$B={\begin{bmatrix}1&0&-1&5\\0&-2&5&-6\\0&0&0&0\end{bmatrix}}.$

(a) List rank  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 \text{dim Nul }A.}

(b) Find bases 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 \text{Col }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 \text{Nul }A.}   Find an example of a nonzero vector that belongs to  ${\text{Col }}A,$  as well as an example of a nonzero vector that belongs to  ${\text{Nul }}A.$

 Problem 2 

Find the dimension of the subspace spanned by the given vectors. Are these vectors linearly independent?

${\begin{bmatrix}1\\0\\2\end{bmatrix}},{\begin{bmatrix}3\\1\\1\end{bmatrix}},{\begin{bmatrix}-2\\-1\\1\end{bmatrix}},{\begin{bmatrix}5\\2\\2\end{bmatrix}}$

 Problem 3 

Let  $B={\begin{bmatrix}1&-2&3&4\\0&3&0&0\\0&5&1&2\\0&-1&3&6\end{bmatrix}}.$

(a) Is  $B$  invertible? Explain.

(b) Define a linear transformation  $T$  by the formula  $T({\vec {x}})=B{\vec {x}}.$  Is  $T$  onto? Explain.

 Problem 4 

Suppose  $T$  is a linear transformation given by the formula

$T{\Bigg (}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\\\end{bmatrix}}{\Bigg )}={\begin{bmatrix}5x_{1}-2.5x_{2}+10x_{3}\\-x_{1}+0.5x_{2}-2x_{3}\end{bmatrix}}$

(a) Find the standard matrix for  $T.$

(b) Let  ${\vec {u}}=7{\vec {e_{1}}}-4{\vec {e_{2}}}.$  Find  $T({\vec {u}}).$

(c) Is  ${\begin{bmatrix}-1\\3\end{bmatrix}}$  in the range of  $T?$  Explain.

 Problem 5 

Let  $A$  and  $B$  be  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 6\times 6}   matrices with  ${\text{det }}A=-10$  and  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{det }}B=5.}   Use properties of determinants to compute:

(a)  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{det }}3A}

(b)  ${\text{det }}(A^{T}B^{-1})$

 Problem 6 

Let  ${\vec {v}}={\begin{bmatrix}-1\\3\\0\end{bmatrix}}$  and  ${\vec {y}}={\begin{bmatrix}2\\0\\5\end{bmatrix}}.$

(a) Find a unit vector in the direction of  ${\vec {v}}.$

(b) Find the distance between  ${\vec {v}}$  and  ${\vec {y}}.$

(c) Let  $L={\text{Span }}\{{\vec {v}}\}.$  Compute the orthogonal projection of  ${\vec {y}}$  onto  $L.$

 Problem 7 

(a) Let  $T:\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$  be a transformation given by

$T{\bigg (}{\begin{bmatrix}x\\y\end{bmatrix}}{\bigg )}={\begin{bmatrix}1-xy\\x+y\end{bmatrix}}.$

Determine whether  $T$  is a linear transformation. Explain.

(b) Let  $A={\begin{bmatrix}1&-3&0\\-4&1&1\end{bmatrix}}$  and  $B={\begin{bmatrix}2&1\\1&0\\-1&1\end{bmatrix}}.$  Find  $AB,~BA^{T}$  and  $A-B^{T}.$

 Problem 8 

Let  $A={\begin{bmatrix}1&3&8\\2&4&11\\1&2&5\end{bmatrix}}.$  Find  $A^{-1}$  if possible.

 Problem 9 

If  $A$  is an  $n\times n$  matrix such that  $AA^{T}=I,$  what are the possible values of  ${\text{det }}A?$

 Problem 10 

(a) Suppose a  $6\times 8$  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}   has 4 pivot columns. What 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 \text{dim Nul }A?}   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 \text{Col }A=\mathbb{R}^4?}   Why or why not?

(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 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 7\times 5}   matrix, what is the smallest possible dimension 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 \text{Nul }A?}

 Problem 11 

Consider the following system of equations.

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 x_1+kx_2=1}

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 3x_1+5x_2=2k}

Find all real values 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 k}   such that the system has only one solution.