Module 4 Quiz: Vector Spaces
Quiz
10 questions on vectors, subspaces, independence, basis, and dimension.
1
(1,2,3) . (4,-1,2) = ?
4-2+6 = 8.
2
Normalize (0,3,4).
||(0,3,4)|| = 5. Unit vector: (0, 3/5, 4/5).
3
Is {(x,y): y = x^2} a subspace of R^2?
No. Not closed under addition. (1,1) and (2,4) are in the set but (3,5) is not (5 is not 9).
4
What is Span{(0,0,0)}?
{(0,0,0)}. Any scalar times the zero vector is still the zero vector.
5
Are (1,0) and (0,1) linearly independent?
Yes. Neither is a scalar multiple of the other. det([1 0;0 1])=1 (nonzero).
6
What is dim(R^7)?
7.
7
A 5x8 matrix has rank 4. What is its nullity?
8 - 4 = 4.
8
Which columns of A form a basis for Col(A)?
The pivot columns of the original matrix A (identified by row reducing).
9
Can 5 vectors in R^3 be linearly independent?
No. In R^3, at most 3 vectors can be independent.
10
State the Rank Theorem.
rank(A) + nullity(A) = n, where n is the number of columns of A.