Difference between revisions of "Section 1.12 Homework"
(Created page with "'''8.''' Let <math>L: V \to W</math> be a linear map.<br /> <br /> (b) Show that if <math>x_1,x_2,...,x_k</math> are linearly dependent, then <math>L(x_1),L(x_2),...,L(x_k)</m...") |
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(b) Show that if <math>x_1,x_2,...,x_k</math> are linearly dependent, then <math>L(x_1),L(x_2),...,L(x_k)</math> are linearly dependent.<br /> | (b) Show that if <math>x_1,x_2,...,x_k</math> are linearly dependent, then <math>L(x_1),L(x_2),...,L(x_k)</math> are linearly dependent.<br /> | ||
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| − | + | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | |
| + | !Proof: | ||
| + | |- | ||
| + | |Suppose that <math>x_1,x_2,...,x_k</math> are linearly dependent. Then there are scalars <math>c_1,c_2,...,c_k</math>, not all of which are zero that satisfy <math>c_1 x_1 + c_2 x_2 + \cdots +c_k x_k = 0</math>. Now recall that for any linear transformation <math>L(0) = 0</math>. So then <math>L(c_1 x_1 + \cdots +c_k x_k) = L(0) = 0</math>. But by linearity of <math>L</math> we have <math>L(c_1x_1 + \cdots +c_k x_k) = L(c_1 x_1) + L(c_2 x_2) + \cdots +L(c_k x_k) = c_1 L(x_1) + c_2 L(x_2) + \cdots +c_k L(x_k)</math>. Combining these facts gives that <math>c_1 L(x_1) + \cdots +c_k L(x_k) = 0</math>. In other words, we have a linear combination of <math>L(x_1),L(x_2),...,L(x_k)</math> that gives zero and we know that not all of the <math>c_1,...,c_k</math> are zero. Therefore <math>L(x_1),L(x_2),...,L(x_k)</math> are linearly dependent.<br /> | ||
| + | |} | ||
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(c) Show that if <math>L(x_1),L(x_2),...,L(x_k)</math> are linearly independent then <math>x_1,x_2,...,x_k</math> are linearly independent.<br /> | (c) Show that if <math>L(x_1),L(x_2),...,L(x_k)</math> are linearly independent then <math>x_1,x_2,...,x_k</math> are linearly independent.<br /> | ||
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| − | Note: This is the contrapositive statement of part (b). Hence since we proved (b), then (c) is also true as contrapositives are logically equivalent. However, we can prove this separately as follows.<br /> | + | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" |
| + | !Proof: | ||
| + | |- | ||
| + | |Note: This is the contrapositive statement of part (b). Hence since we proved (b), then (c) is also true as contrapositives are logically equivalent. However, we can prove this separately as follows.<br /> | ||
<br /> | <br /> | ||
''Proof:'' Suppose that <math>L(x_1),L(x_2),...,L(x_k)</math> are linearly independent. To show that <math>x_1,x_2,...,x_k</math> are linearly independent we consider any combination <math>c_1 x_1 + c_2 x_2 + \cdots +c_k x_k</math> that gives 0. We want to show that this can only happen if all of <math>c_1,c_2,...,c_k =0</math>. Since <math>c_1 x_1 + c_2 x_2 + \cdots +c_k x_k = 0</math>, then <math>L(c_1 x_1 + \cdots +c_k x_k) = L(0) = 0</math>. As in the proof of part (b) we then have <math>c_1 L(x_1) + c_2 L(x_2) + \cdots +c_k L(x_k) = 0</math>. That is, we have found a linear combination of <math>L(x_1),L(x_2),...,L(x_k)</math> that gives zero. But since <math>L(x_1),L(x_2),...,L(x_k)</math> are linearly independent, then we must have <math>c_1 = c_2 = \cdots = c_k = 0</math>. Therefore <math>x_1,x_2,...,x_k</math> are linearly independent. | ''Proof:'' Suppose that <math>L(x_1),L(x_2),...,L(x_k)</math> are linearly independent. To show that <math>x_1,x_2,...,x_k</math> are linearly independent we consider any combination <math>c_1 x_1 + c_2 x_2 + \cdots +c_k x_k</math> that gives 0. We want to show that this can only happen if all of <math>c_1,c_2,...,c_k =0</math>. Since <math>c_1 x_1 + c_2 x_2 + \cdots +c_k x_k = 0</math>, then <math>L(c_1 x_1 + \cdots +c_k x_k) = L(0) = 0</math>. As in the proof of part (b) we then have <math>c_1 L(x_1) + c_2 L(x_2) + \cdots +c_k L(x_k) = 0</math>. That is, we have found a linear combination of <math>L(x_1),L(x_2),...,L(x_k)</math> that gives zero. But since <math>L(x_1),L(x_2),...,L(x_k)</math> are linearly independent, then we must have <math>c_1 = c_2 = \cdots = c_k = 0</math>. Therefore <math>x_1,x_2,...,x_k</math> are linearly independent. | ||
| + | |} | ||
Latest revision as of 23:06, 15 November 2015
8. 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 L: V \to W}
be a linear map.
(b) Show that 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 x_1,x_2,...,x_k}
are linearly dependent, 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 L(x_1),L(x_2),...,L(x_k)}
are linearly dependent.
| Proof: |
|---|
| Suppose 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 x_1,x_2,...,x_k}
are linearly dependent. Then there are scalars 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_1,c_2,...,c_k}
, not all of which are zero that satisfy 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_1 x_1 + c_2 x_2 + \cdots +c_k x_k = 0}
. Now recall that for any linear transformation 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 L(0) = 0}
. So 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 L(c_1 x_1 + \cdots +c_k x_k) = L(0) = 0}
. But by linearity 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 L}
we have 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 L(c_1x_1 + \cdots +c_k x_k) = L(c_1 x_1) + L(c_2 x_2) + \cdots +L(c_k x_k) = c_1 L(x_1) + c_2 L(x_2) + \cdots +c_k L(x_k)}
. Combining these facts gives 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 c_1 L(x_1) + \cdots +c_k L(x_k) = 0}
. In other words, we have a linear combination 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 L(x_1),L(x_2),...,L(x_k)}
that gives zero and we know that not all of the 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_1,...,c_k}
are zero. Therefore 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 L(x_1),L(x_2),...,L(x_k)}
are linearly dependent. |
(c) Show that 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 L(x_1),L(x_2),...,L(x_k)}
are linearly independent 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 x_1,x_2,...,x_k}
are linearly independent.
| Proof: |
|---|
| Note: This is the contrapositive statement of part (b). Hence since we proved (b), then (c) is also true as contrapositives are logically equivalent. However, we can prove this separately as follows.
|