Math 18 Week 7 Jeopardy

Invertible Matrix Theorem

100

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.

200

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.

300

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.

400

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

500

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.

Vector Spaces 1

100

T/F: The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of R^m.

False, Null space of an mxn matrix is a subspace of Rⁿ

200

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.

300

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).

400

If a 6x3 matrix A has rank 3, find dimNul(A) and rank A^T.

dimNul(A) = 0, 

dimCol(A^T) = rank(A^T) = 3

500

Suppose A is mxn and b is in R^m. 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.

Vector Spaces 2

100

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.

200

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

300

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.

400

T/F: If there exists a set {v_{1, ...,}  v_p} that spans V, then dim V is less than or equal to p. 

True, a spanning set for a vector space includes its basis.

500

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.

Determinants

100

T/F: A square matrix A in invertible if det(A) = 0

False

200

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

300

Compute the determinant of the following matrix using cofactor expansion

See PPT

400

A matrix with only zeros above its main diagonal is called _____ triangular

Lower

500

If det(A) = 2, what is det(A³)?

det(A³) = det(A*A*A) = det(A)*det*(A)*det(A) = 2*2*2 = 8