Direct Sums And Inverses

Question 1

What is direct sum of vector spaces U and W

Answer 1

Direct sum of vector spaces U and W is:
For all u, w in U and W,
(U1,W1) + (U2,W2) = (U1+U2,W1+W2)
L(u,w) = (Lu,Lw)

Question 2

What is dim(U +x W)

Answer 2

Dim(U +x W) = dimU + Dim W

Question 3

What is dim(U+W) + dim(U N W)

Answer 3

Dim(U+W) + dim(U N W) = dimU + dimW

Question 4

What is Sn

Answer 4

Sn is set of objections/permutations sigma from I to I (set o n elements)

Question 5

What is det of block wise upper triangular matrix

Answer 5

Det of blockwise upper triangular matrix is:
Det of top left block * det of bottom right block
Blocks all have 0s underneath

Question 6

What Does det(LA) =

Answer 6

Det(LA) = L^n det(A) (nxn matrix)

Question 7

What are properties of determinant and when does this also hold

Answer 7

Properties of determinant are:
Mulitlinear
Alternating
This also holds when viewing det A as a function of rows instead of columns as det A = det A^T

Question 8

What are definitions of multilinear and alternating

Answer 8

Definition of multilinear is:
If Cj = Lv + Uw, v,w vectors L,U scalars
Det(C1,…,Cj,…,Cn) = Ldet(C1,…,v,…,Cn) + Uder(C1,…,w,…Cn) where Cj denotes a column
Definition of alternating is:
If Ci =Cj I=/j, then det(C1,…,Cn) = 0
These properties determine how det changes under ECOs

Question 9

If B is obtained from A by Bci= LCi then what is det B

Answer 9

If B is obtained from A by Bi = LCi then det B = Ldet A

Question 10

If B is obtained from A by BCi = Ci + LCj, what is det B

Answer 10

If B is obtained from A by BCi = Ci + LCj, then det B = det A

Question 11

If B is obtained from A by swapping 2 columns, what is det B

Answer 11

If B is obtained from A by swapping 2 columns det B = -det A

All previous properties can be applied with rows as det A = det A^T