Bases
T/F: The columns of an invertible nxn matrix form a basis for Rn.
True, an invertible nxn matrix has n linearly independent columns, which can form a basis for Rn.
det(A+B)=detA+detB
False. only works with det(AB)=det(A)*det(B)
T/F: The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of Rm.
False, Null space of an mxn matrix is a subspace of Rn
T/F: Let U be a subspace of R5. dim(U) ≤ 5.
True, a basis for R5 has at most 5 vectors.
If x is in V and if B contains n vectors, then the B-coordinate vector of x is in Rn
True - definition
Is B = {(1, 0, 0), (1, 1, 0), (1, 1, 1)} a basis for R3?
Yes, B is a linearly independent spanning set of 3 vectors in R3.
If det(A) = 4, what is det(A^3)?
det(A^3) = det(A*A*A) = det(A)*det*(A)*det(A) = 4*4*4 = 64
A = [3 0 2;
6 0 4;
0 0 0]
Is v = (-2, 2, 3) in Nul(A)?
Yes, Av = 0.
T/F: If dim(V) = p, then there exists a spanning set of p+1 vectors in V.
True, the additional vector can be linearly dependent to any vector in the basis, or we could even use the zero vector.
The vector spaces P3 and R3 are isomorphic.
False - isomorphic to R4
What is a matrix with only zeros above its main diagonal called and how would you compute its determinant.
Lower Triangular Matrix, Product of diagonal entries
Let A = [-6 12;
-3 6 ]
and w = (2, 1).
Is w in Col(A)?
Yes, Ax = w is consistent
If a 3x5 matrix A has rank(A) = 3, find dimNul(A) and rank(AT).
dimNul(A) = 2,
rank(AT) = 3
Let A be a 3x3 matrix with det(A) = 2 and B = 3A. What is det(B)?
det(B) = det(3A) = (3^3)*det(A) = 27*2 = 54
H = {(x,y,z) : 2x+y=z, z=-x, x=3y-z}. Is H a subspace of R3? Justify your reasoning
H is a subspace. H = Nul(A) where A =
[2 1 -1;
1 0 1;
1 -3 1;]
T/F: If there exists a set {v1, ..., vp} that spans V, then dim V is less than or equal to p.
True - spanning set theorem
Use coordinate vectors to test the linear independence of the set of polynomials
1+2t3
2+t-3t2
-t+2t2-t3
Since the matrix has a pivot in each column, its columns (and thus the given polynomials) are linearly independent.
Compute the determinant of the following matrix using cofactor expansion
[ 1 3 5;
2 0 -1;
4 -3 1 ]
-51
W = {(4a+3b, 0, a+b+c, c-2a) : a,b,c are real numbers}
Is W a subspace? Justify your reasoning.
W is a subspace.
W = span{(4, 0, 1, -2), (3, 0, 1, 0), (0, 0, 1, 1)}
Suppose A is mxn and b is in Rm. What has to be true about the two numbers rank[A|b] and rank(A) in order for the equation Ax=b to be consistent?
rank[A|b] = rank(A) because we want b to be a linear combination of the columns of A.