Homework Answers
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5 3 1 0 Problem 10 Let wi = ,W2 W3 Let W = Span{W1,W2, W3}...
5 3 1 0 Problem 10 Let wi = ,W2 W3 Let W = Span{W1,W2, W3} C R6. 11 9 1 2 a) [6 pts] Use the Gram-Schmit algorithm to find an orthogonal basis for W. You should explicitly show each step of your calculation. 10 -7 11 b) [5 pts) Let v = Compute the projection prw(v) of v onto the subspace W using the 5 orthogonal basis in a). c) (4 pts] Use the computation in b) to...
2. Let M2x2(R) be the vector space consisting of 2 x 2 matrices with real entries....
2. Let M2x2(R) be the vector space consisting of 2 x 2 matrices with real entries. Let W M2x2 (R) det (A) 0. Show that W is not a subspace of M2x2(R) A E
3. Let V be the subspace of M2x2(R) consisting of all matrices in which the sum...
3. Let V be the subspace of M2x2(R) consisting of all matrices in which the sum of entries on each row is equal to 0. Let W be the subspace of M2x2(R) consisting of all matrices in which the sum of entries on each column is equal to 0. Find a basis of V +W.
Question 7 (6 marks) Consider M2x2 equipped with the inner product Let S c M2x2 be...
Question 7 (6 marks) Consider M2x2 equipped with the inner product Let S c M2x2 be the subspace spanned by [1 21 [0 1 (a) Find an orthonormal basis for S (b) Find the element of S closest to 0 1
2. Let Wi-((a, b, c) : a-c-b), W2-((a, b, c) : ab>0), W3-((z, y,z) : r2+92+22£1} be subsets of R3 (a) Determine which of these subsets is a subspace of R3. Justify your answer. (b) For the subsets...
2. Let Wi-((a, b, c) : a-c-b), W2-((a, b, c) : ab>0), W3-((z, y,z) : r2+92+22£1} be subsets of R3 (a) Determine which of these subsets is a subspace of R3. Justify your answer. (b) For the subsets which are subspaces, find a basis and the dimension for each of them
2. Let Wi-((a, b, c) : a-c-b), W2-((a, b, c) : ab>0), W3-((z, y,z) : r2+92+22£1} be subsets of R3 (a) Determine which of these subsets is a subspace...
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be the set of...
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be
the set of all 2x2 skew-symmetric matrices: W = {A in m2x2(R) l
A^T=-A}.
(a) Show that W is a subspace of M2x2(R)
(b) Find a basis for W and determine dim(W).
(c) Suppose T: M2x2(R) is a linear transformation given by
T(A)=A^T +A. Is T injective? Is T surjective? Why or why not? You
do not need to verify that T is linear.
3. (17 points)...
Let W be the subspace spanned by the given vectors. Find a basis for Wt, 0...
Let W be the subspace spanned by the given vectors. Find a basis for Wt, 0 1 A. W1 = W2 = 3 2 -1 2 B. W W2 2 -3 W3 = 6
ote: The norm of is denoted by |vand is calculated N a vector u Consider a subspace W of R4, W span(1, v2, v3, v4)). Where 0 из- 1. Find an orthonormal basis Qw of W and find the dimension of W 2. Fi...
ote: The norm of is denoted by |vand is calculated N a vector u Consider a subspace W of R4, W span(1, v2, v3, v4)). Where 0 из- 1. Find an orthonormal basis Qw of W and find the dimension of W 2. Find an orthonormal basis QWL of WL and find the dimension of WL 3. GIven a vector u- . find the Qw coordinate of Projw(v) . find the Qwa coordinate of Projwi (v) » find the coordinate...
D. Let W, - W, n W,. Find a basis for W, (but don't prove it). What is the dimension of w? ld J ...
d. Let W, - W, n W,. Find a basis for W, (but don't prove it). What is the dimension of w? ld J matrices. the subspace of symmetric 3x3 W, A in M: A" -A : Ws = the subspace of 3x3 matrices having the property that the sum of its entries is zero
d. Let W, - W, n W,. Find a basis for W, (but don't prove it). What is the dimension of w? ld J
matrices....
Let W be the set of singular (noninvertible) matrices of order 2. Show that W is...
Let W be the set of singular (noninvertible) matrices
of order 2. Show that W is not a subspace of
M2×2 with the standard matrix operations.
Q1: Let W be the set of singular (noninvertible) matrices of order 2. Show that W is not a subspace of M2x2 with the standard matrix operations.-
5 3 1 0 Problem 10 Let wi = ,W2 W3 Let W = Span{W1,W2, W3} C R6. 11 9 1 2 a) [6 pts] Use the Gram-Schmit algorithm to find an orthogonal basis for W. You should explicitly show each step of your calculation. 10 -7 11 b) [5 pts) Let v = Compute the projection prw(v) of v onto the subspace W using the 5 orthogonal basis in a). c) (4 pts] Use the computation in b) to...
2. Let M2x2(R) be the vector space consisting of 2 x 2 matrices with real entries. Let W M2x2 (R) det (A) 0. Show that W is not a subspace of M2x2(R) A E
3. Let V be the subspace of M2x2(R) consisting of all matrices in which the sum of entries on each row is equal to 0. Let W be the subspace of M2x2(R) consisting of all matrices in which the sum of entries on each column is equal to 0. Find a basis of V +W.
Question 7 (6 marks) Consider M2x2 equipped with the inner product Let S c M2x2 be the subspace spanned by [1 21 [0 1 (a) Find an orthonormal basis for S (b) Find the element of S closest to 0 1
2. Let Wi-((a, b, c) : a-c-b), W2-((a, b, c) : ab>0), W3-((z, y,z) : r2+92+22£1} be subsets of R3 (a) Determine which of these subsets is a subspace of R3. Justify your answer. (b) For the subsets which are subspaces, find a basis and the dimension for each of them
2. Let Wi-((a, b, c) : a-c-b), W2-((a, b, c) : ab>0), W3-((z, y,z) : r2+92+22£1} be subsets of R3 (a) Determine which of these subsets is a subspace...
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be
the set of all 2x2 skew-symmetric matrices: W = {A in m2x2(R) l
A^T=-A}.
(a) Show that W is a subspace of M2x2(R)
(b) Find a basis for W and determine dim(W).
(c) Suppose T: M2x2(R) is a linear transformation given by
T(A)=A^T +A. Is T injective? Is T surjective? Why or why not? You
do not need to verify that T is linear.
3. (17 points)...
Let W be the subspace spanned by the given vectors. Find a basis for Wt, 0 1 A. W1 = W2 = 3 2 -1 2 B. W W2 2 -3 W3 = 6
ote: The norm of is denoted by |vand is calculated N a vector u Consider a subspace W of R4, W span(1, v2, v3, v4)). Where 0 из- 1. Find an orthonormal basis Qw of W and find the dimension of W 2. Find an orthonormal basis QWL of WL and find the dimension of WL 3. GIven a vector u- . find the Qw coordinate of Projw(v) . find the Qwa coordinate of Projwi (v) » find the coordinate...
d. Let W, - W, n W,. Find a basis for W, (but don't prove it). What is the dimension of w? ld J matrices. the subspace of symmetric 3x3 W, A in M: A" -A : Ws = the subspace of 3x3 matrices having the property that the sum of its entries is zero
d. Let W, - W, n W,. Find a basis for W, (but don't prove it). What is the dimension of w? ld J
matrices....
Let W be the set of singular (noninvertible) matrices
of order 2. Show that W is not a subspace of
M2×2 with the standard matrix operations.
Q1: Let W be the set of singular (noninvertible) matrices of order 2. Show that W is not a subspace of M2x2 with the standard matrix operations.-
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