10. Find the coordinate vector of p relative to the basis S = {p1, p2, p3}.

(a) p = 4 - 3x + x2; p1 = 1, p2 = x, p3 = x2
(b) p = 2 - x + x2; p1 = 1 + x, p2 = 1 + x2, p3 = x + x2

22. Find the standard basis vectors that can be added to the set {v1, v2} to produce a basis for R4.

v1 = (1, -4, 2, -3), v2 = (-3, 8, -4, 6)

30. Prove: Any subspace of a finite-dimensional vector space is finite-dimensional.
Hint: You can do a proof by contradiction by using 2 or 3 Theorems from section.

10. Find the coordinate vector of p relative to the basis S = {p1, p2, p3}.

The coordinates of p relative to the basis S are the numbers c1, c2, c3 such that p = c1p1 + c2p2 + c3p3. The coordinate vector of p would be the vector {c1, c2, c3}. (Note how this relates to theorem 5.4.1 - there is only one coordinate vector for each p.)

(a) p = 4 - 3x + x2; p1 = 1, p2 = x, p3 = x2

p = c1(p1) + c2(p2) + c3(p3)
p = c1(1) + c2(x) + c3(x2)

c1 = 4, c2 = -3, c3 = 1

(b) p = 2 - x + x2; p1 = 1 + x, p2 = 1 + x2, p3 = x + x2

p = c1(p1) + c2(p2) + c3(p3)
p = c1(1 + x) + c2(1 + x2) + c3(x + x2)

2. Let T and S be matrix multiplication transformations from R^2 into R^3, described as
T[x, y] = [1,2; 1,1][x,y] and S[x,y] = [7,4; 6,7][x,y]
Find the transformations 2T - S, ST and TS. Do T and S commute?
3. Let U = {all[a,c; 5a,3c]},
that is U is the set of all 2 x 2 matrices A such that a(12) = 5a(11), a(22) = 3a(23)

I have two questions that I need help with.
1) How would you find a basis of the kernel, a basis of the image and determine the dimension of each for this matrix? The
matrix is in the attachment.
2) Are the following 3 vectors linearly dependent? (see attachment for the three vectors) How can you decide?
I hope y

Hi, I need some assistance with all the attached questions. I am not too sure how to answer them and a step-by-step working guide for each question would really help me in understanding these problems. These are all linear algebra problems.

Find the dimensions of each of the following vector spaces.
a) The vector space of all diagonal n X n matrices
b) The vector space of all symmetric n X n matrices
c) The vector space of all upper triangular n X n matrices

Let T be a linear map on R^2 defined by T(x,y) = (4x - 2y, 2x + y).
Calculate the matrix of T relative to the basis {α1, α2} where α1 = (1,1) , α2 = (-1,0).

Could you clarify what constitutes a spanning set and a basis? Also how does one test to see if a set of vectors is a spanning set and if it is a basis?

Question (1)
Find a basisanddimension of the subspace W of R4 generated by the vectors
( 1 , - 4 , - 2 , 1 ) , ( 1 , - 3 , - 1 , 2 ) , ( 3 , - 8 , - 2 , 7 ) .
Extend it to find the basis of R4 .
Question (2)
Determine a basisand the dimensions of the Subspace of M2(R) generated by the
2 by 2 Matrices [ 2 -10 ] , [

I need the proof of the Linear programming problem attached.
---
Consider the LP:
Min ct x
Subject to
Ax ≥ b, x ≥ 0.
One can convert the problem to an equivalent one with equality constraints by using slack variables. Suppose that the optimal basis for the equality constrained problem is B. Prove t

Question (1)
Find a basisanddimension of the subspace W of R4 generated by the vectors
( 1 , - 4 , - 2 , 1 ) , ( 1 , - 3 , - 1 , 2 ) , ( 3 , - 8 , - 2 , 7 ) .
Extend it to find the basis of R4
Question (2)
Determine a basisand the dimensions of the Subspace of M2(R) generated by the
2 by 2 Matrices [ 2 -10 ] , [

1) Let { 1, 2, 2........... n} be a basis of an n dimensional vector space over R and A be n Matrix .
Let ( 1, 2, 3............... s) = ( 1, 2, 2........... n) A
Prove that dim (span { 1, 2, 3............... s}) = Rank (A).
2) Let V1 be the solution space of x1 +x2 + x3............+xn = 0
let V2 be the solution spac