Find the distance between the vectors
u = [1 0 1]
v = [2 2 2]
sqrt(6)
Is the matrix
[1 2
5 10]
invertible?
No! Its determinant is 0.
Which vector must always be in any subspace of a vector space?
The zero vector
Find the determinant of
[1 2 3
0 4 5
0 4 5]
0
Define what it means for a vector to be an eigenvector of a matrix A
There is a nonzero vector x so that
Ax = (lambda)x
for some scalar (lambda)
Write [1 2 3] as a linear combination of
u = [1 0 1]
v = [0 1 1]
w = [-1 0 1]
1u + 2v +0w
Is the matrix
[1 0 1 2 3
0 1 2 0 1
0 0 0 1 0]
in reduced row echelon form?
No!
There is a 2 above the third pivot
What is the definition of the Null Space of a matrix A?
The set of solutions to Ax=0.
The determinant of
[-1 2 3
0 9 9
0 0 3]
-27
What are the eigenvalues of the matrix
[2 -3
3 2]?
2 + 3i, 2-3i
Are the vectors u,v,w linearly independent?
v = [1 2 3]
w = [4 5 6]
u = [7 8 9]
No! The determinant of
[1 4 7
2 5 8
3 6 9]
is 0, so the columns are not linearly independent.
If A is 5 x 3 and B is 3 x 6, what are the dimensions of AB and BA?
AB is 5x6, BA does not exist
Is the vector u = [1 2 3]
in the column space of
[1 0 0
0 2 1
1 0 0]?
No! The augmented matrix
[1 0 0 1
0 2 1 2
1 0 0 3]
represents a system with no solution.
Find the area of the parallelogram with corners given by
(0,0), (1,1), (2,3), (3, 4)
1
Find the eigenvalues and eigenvectors of the matrix
[3 0 1
0 2 3
0 0 4]
3, 2, 4
[1,0,0]
[0,1,0]
[1,3/2,1]
Consider the basis
u_1 = [1 2 3]
u_2 = [1 0 1]
u_3 = [0 1 0]
If x has coordinate vector [4 3 2] find x.
[7 10 15]
Find the inverse of the matrix
[1 0 1
0 1 0
0 2 1]
[1 2 -1
0 1 0
0 -2 1]
Find a basis of the column space of
[1 2 3
4 5 6
7 8 9]
and find its dimension.
Basis: [1 4 7], [2 5 8]
Dimension: 2
Use Cramer's rule to solve
3x + 3y = 17
x - 2y = 13
x = 73/9
y = -22/9
Diagonalize the matrix
[2 1 0
0 3 0
0 0 3]
D = diag(2, 3, 3)
P =
[1 0 1
0 0 1
0 1 0]
u = [1 2 3]
onto the line spanned by v
v = [3 2 1]
projection is:
5/7 [3 2 1]
OR
[15/7, 10/7, 5/7]
T(x_1, x_2, x_3) = (x_1 + x_2, x_2 + x_3, x_3).
Is there a vector whose image under T is (1, 1, 1)?
Find that vector if so, or explain why not.
Yes, we find that
(x_1, x_2, x_3) = (1, 0, 1)
Is it possible for the rank of a 5 x 7 matrix A to be 4? If so, what must be true about the null space of A?
Yes! Rank = 4 means A has 4 pivots.
In this case, A has 3 non-pivot columns, so Ax=0 will have 3 free variables, and thus the dimension of the null space is 3 (i.e. the nullity of A is 3).
A linear transformation transforms an image by shrinking its area by a factor of 1/4 and not distorting the image in any direction. What is the linear transformation?
T(x) = Ax where A =
[1/2 0
0 1/2]
The matrix
[-1 -1
2 1]
is similar to a matrix with rotates and rescales vectors. Find that matrix and the angle by which it rotates vectors
lambda = i, -i
matrix =
[0 -1
1 0]
Rotation by pi/2 radians (90 degrees)