Invertible Matrix Theorem
T/F: If the linear transformation T(x)=Ax maps Rn to Rn, then A has n pivot positions.
False, we don't know enough about the transformation.
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,V vector spaces be subsets of R5. The set S = {u : u is a vector in U OR vector in V} is a subspace for R5.
False, does not satisfy u+v, u in U and v in V.
T/F: A square matrix A in invertible if det(A) = 0
False
If A is an nxn matrix and the transformation T(x)=Ax is one-to-one, then the transformation is also onto.
True, the transformation is also onto by the Invertible Matrix Theorem.
T/F: The columns of an invertible nxn matrix form a basis for Rn.
True, linearly independent vectors can be row reduced to standard basis vectors.
T/F: Let U a vector space be a subset of R5. dimU is less than or equal to 5.
True, a basis for R5 has at most 5 vectors
Let A be a 3x3 matrix with det(A) = 2 and B = 3A. What is det(B)?
det(B) = det(3A) = (33)*det(A) = 27*2 = 54
Let A be a 6x4 matrix and B a 4x6 matrix. Show that the 6x6 matrix AB cannot be invertible. (Hint: consider the equation ABx = 0)
Since B is 4x6 (more columns than rows), Bx = 0 has a nontrivial solution x. Then, ABx = 0 also has a nontrivial solution and thus is not invertible.
T/F: If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col(A).
False, we use the columns of A to form the basis for Col(A).
T/F: A plane in R3 that passes through the origin is a two dimensional subspace of R3.
True, check that is satisfies all three properties of subspace.
Compute the determinant of the following matrix using cofactor expansion
See PPT
Suppose (B-C)*D=0, where B and C are mxn matrices and D is invertible. Show that B = C.
(B-C)*D*D-1=0*D-1
(B-C)=0
B=C
If a 6x3 matrix A has rank 3, find dimNul(A) and rank AT.
dimNul(A) = 0,
dimCol(AT) = rank(AT) = 3
T/F: If there exists a set {v1, ..., vp} that spans V, then dim V is less than or equal to p.
True, a spanning set for a vector space includes its basis.
A matrix with only zeros above its main diagonal is called _____ triangular
Lower
Explain why the columns of A2 span Rn whenever the columns of A are linearly independent.
If A2 exists, A must be a square matrix. Since A is square and has linearly independent columns, A is invertible. Then also A2 is invertible, and by the Invertible Matrix Theorem, the columns of A2 span Rn.
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.
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.
If det(A) = 2, what is det(A3)?
det(A3) = det(A*A*A) = det(A)*det*(A)*det(A) = 2*2*2 = 8