APMA 3080 Test 2 Material

Requirements to be a subspace

- closed under addition
- closed under scalar multiplication
- the 0 vector must be in S
- S is a subspace if S contains all linear combinations of the vectors in S

Range

Span{columns of A}

Ker(T) = ___(A)?

Ker(T) = null(A)

Rank Nullity Theorem

Rank(A) + Nullity(A) = total # of columns in A

Rank(A)

Dim(col(A))
-# of leading 1s in rref(A)

Nullity(A)

dim(null(A))
- # of nonleading 1s in RREF(A)

When will a matrix not be invertible?

If the determinant is 0.

Rewrite Det(A^2)

(Det(A))^2

Det(AB)

Det(A)Det(B)

If A has a row or column of 0's, what happens to the determinant?

Det = 0

If two distinct rows (or columns) of a matrix are interchanged, what happens to the determinant?

makes the original determinant negative

If one row (or column) is multiplied by a constant u, what happens to the determinant?

udet(A)

If two distinct rows (or columns) of A are identical, what's the determinant?

det(A) = 0

If a multiple of one row is added to a different row, what's the determinant?

Stays the same -- no difference

If an entire matrix is multiplied by a scalar c, what happens to the determinant?

(c)^n*det(A)

What is the determinant of A^t?

Det(A)

When given a characteristic polynomial, what's the size of A?

The sum of the multiplicities of all of the eigenvalues (will always be a square matrix)

Trace(A)

Sum of all of the eigenvalues

Determinant of A is equal to what, relating to eigenvalues?

The product of the eigenvalues

What is the largest possible dimension for the eigenspace of A?

R^(highest multiplicity of one of the eigenvalues)

Det(cA) = ...

c^n(det(A))

What does raising a matrix to a power due to the eigenvalue

It raises the eigenvalue to that same power

Inverting a matrix does what to the eigenvalue

Inverts the eigenvalue

Subtracting any multiple of the identity matrix does what to the eigenvalue

Also subtracts that value from the eigenvalue

Multiplying a matrix by a scalar does what to the associated eigenvalues?

Multiplies the eigenvalues by the same scalar