UWthat L(Jf(x))= f(x) f(0) is a linear operator in C[-1,1.(b) Find ker L above.Find the range of L above.10. Let S = {(x1,x2, x3, 4) 1+2x3 +4} be a subspace of R4. Find S-.11. Given v =(1,-1, 1, 1) and w =(4, 2,2,1)(a) Determine the angle between v and w.bFind the orthogonal complement of Vspan fv, w}.12. Let A be an m x n matrix.(a) Suppose that rank A = r, what are dimensions of N(A) and N(A).Verify that N(AT A) = N(A) and rank(AT A) =r.

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Asked Nov 11, 2019
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#12 a,b

UWthat L(Jf(x))= f(x) f(0) is a linear operator in C[-1,1.
(b) Find ker L above.
Find the range of L above.
10. Let S = {(x1,x2, x3, 4) 1+2
x3 +4} be a subspace of R4. Find S-.
11. Given v =(1,-1, 1, 1) and w =
(4, 2,2,1)
(a) Determine the angle between v and w.
b
Find the orthogonal complement of V
span fv, w}.
12. Let A be an m x n matrix.
(a) Suppose that rank A = r, what are dimensions of N(A) and N(A).
Verify that N(AT A) = N(A) and rank(AT A) =r.
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UWthat L(Jf(x))= f(x) f(0) is a linear operator in C[-1,1. (b) Find ker L above. Find the range of L above. 10. Let S = {(x1,x2, x3, 4) 1+2 x3 +4} be a subspace of R4. Find S-. 11. Given v =(1,-1, 1, 1) and w = (4, 2,2,1) (a) Determine the angle between v and w. b Find the orthogonal complement of V span fv, w}. 12. Let A be an m x n matrix. (a) Suppose that rank A = r, what are dimensions of N(A) and N(A). Verify that N(AT A) = N(A) and rank(AT A) =r.

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Expert Answer

Step 1

Given

Ais mxnmatrix and rank ( A) =r
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Ais mxnmatrix and rank ( A) =r

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Step 2

Concept use

Rank nullity theorem

Let A be a mxn matrix
rank A r
then
rank(A)+nullity(A)= n ...(i)
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Let A be a mxn matrix rank A r then rank(A)+nullity(A)= n ...(i)

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Step 3

(a) Here...

Dimension of [A]is mxn
Dimension of [A] is nxm
For A
r+N(A) n.from i)
N(A) n-r
For [A]
r+N(Am.from i)
N(A
= m -r
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Dimension of [A]is mxn Dimension of [A] is nxm For A r+N(A) n.from i) N(A) n-r For [A] r+N(Am.from i) N(A = m -r

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