Can someone help me with all the questions? I’m really struggling 2. Find all va

Question

Can someone help me with all the questions? I’m really struggling 2. Find all values of c such that the matrix is NOT invertible. c+4 -1 c+5 -3 3 Lc+6-3 +7 3. Describe the additive identity and additive inverse of the vector space M2s 4. Let A be the standard matrix of a linear transformation T:-. If the range of T has dimension 2, determine the dimension of each of the following subspaces: a) Col A b) Null A c) Row A 5. Determine if the given set is a subspace of : 144 6. Find the basis for a) the null space of A, b) the column space of A, c) the row space of A. 1 2 -1 1 4 -5 7. Find a basis for and the dimension of the solution space of Ax-0 2x, +4x2 3x,-6x, -0 3x, +6x+5x,-11x,-0 a) Prove that ß is a basis for R' b) Find vif [v],# 1-3 c) Find [w), if w[-2

Explanation / Answer

2. Let A =

c+4

-1

c+5

-3

3

-4

c+6

-3

c+7

Then det(A) = -2c -6 = 0 if c = 3. Thus A is not invertible if c = 3.

3. The additive identity of M2x5 is the 2x5 zero matrix, with all its entries equal to 0. The additive inverse of a 2x5 matrix A, whose ijth entry is aij, is the 2x5 matrix –A, whose ijth entry is -aij.

4. The standard matrix A of T: R5 R7 is a 7x5 matrix.

a). Range (T) is same as col(A). Hence dim(col(A)) = 2

b). As per the rank-nullity theorem, dim(null(A)) = 5-2 = 3.

c). dim(Row (A)) =- dim(col(A)) = 2.

5. Let X =(u1, u2,u3,u4)T and Y = (v1, v2,v3,v4)T are 2 arbitrary vectors in S, then X+Y = (u1+v1,u2+v2,u3+v3,u4+v4). Further, u2 = 0, v2 = 0, u4= 0 and v4= 0. Hence X+Y = (u1+v1,0, u3+v3,0)T. However, (u1+v1)2 = u12 +v12 + 2u1 v1 = u33 + v33 +2u1 v1 (u1+v1)3. Hence X+Y S so that S is not closed under vector addition. Hence S is not a subspace of R4.

Please post the remaining questions again, maximum of 4 at a time.

c+4

-1

c+5

-3

3

-4

c+6

-3

c+7

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