640:250 Final T/F (2.6,3.1,3.2,4.1-4.3,5.1-5.3,6.1-6.3,6.5,6.6)

For any m ×n matrix A, the dimension of the null space of A plus the dimension of the column space of A equals

False, the dimension of the null space of any in x n matrix A plus the dimension of its column space equals rank A + nullity A = rankA + (n — rankA) = n.

The dot product of two vectors in Rⁿ is a vector in R^n

False, the dot product of two vectors is a scalar.

The eigenspace of an n ×n matrix A corresponding to an eigenvalue λ is the column space of A−λIdentity(n)

False, the eigenspace of A corresponding to eigenvalue λ is the null space of A−λ*Identity(n)

If v is an eigenvector of a matrix A, then cv is also an eigenvector for any scalar

False, the exception is c = 0

Every n ×n matrix has an eigenvector in R^n

False, the rotation matrix A 90 degrees has no eigenvectors in R²

A vector v is in Col A if and only if Ax=v is consistent

True

A vector v is in Null A if and only if Av=0.

True

A vector v is in Row A if and only if A^Tx=v is consistent

True

Every nonzero subspace of R^n has an orthogonal basis

True

For any matrix A, the row space of A^T equals the column space of A

True

For any nonzero vector v, v *( 1 / (||v||) ) is a unit vector

True

For any subspace W of Rⁿ, the only vector in both W and W⊥ is 0

True

If P and Q are n × n orthogonal matrices, then (PQ)^T is an orthogonal matrix

True

If V is a subspace of dimension k, then every generating set for V contains at least k vectors

True

If a rotation of the x- and y-axes is used to write the equation ax^2 +2bxy +cy^2 =d as a'(x')2 +b'(y'2 =d, then the scalars a' and b' are eigenvalues of {{a,b},{c,d}}

True

If some row of a square matrix consists only of zero entries, then the determinant of the matrix equals zero

True

If v is an eigenvector of a matrix A, then cv is also an eigenvector for any nonzero scalar c.

True

If w is the orthogonal projection of u on a line through the origin of R², then u−w is orthogonal to every vector on the line.

True

Let W be a subspace of Rn. If {w₁,w₂,...,w(subscript k)} is an orthonormal basis for W and {z1,z2,...,z(subscript m)} is an orthonormal basis for W⊥, then {w₁,w₂,...,wk,z₁,z₂,...,z(subscript m)} is an orthonormal basis for Rⁿ.

True

The determinant of any square matrix can be evaluated by a cofactor expansion along any column

True

The row space of any matrix equals the row space of its reduced row echelon form

True

The vectors in the vector form of the general solution of Ax=0 form a basis for the null space of A

True

If A is a 5×5 matrix, then det (−A)=detA

False, if A is a 5 x 5 matrix, then det(—A) = —detA

If A is any square matrix and c is any scalar, then detcA=c detA

False, if A is an n x n matrix, then detcA = (c^n)*detA.

If the determinant of a 2×2 matrix equals zero, then the matrix is invertible

False, if the determinant of a 2 x 2 matrix is nonzero, then the matrix is invertible.

An LU decomposition of every matrix is unique

False, is A is the m x n zero matrix and U = A, then A = L, where U is any mxm unit lower triangular matrix.

A square zero matrix has no eigenvalues

False, it has the eigenvalue 0

Eigenvectors of a matrix that correspond to distinct eigenvalues are orthogonal.

False, let A = {{1,4},{1,1}}. Then {2,1} and {2,-1} are eigenvectors of A that correspond to the eigenvalues 3 and -1, respectively. But these two eigenvectors are not orthogonal.

Every subspace of R^n has a basis composed of standard vector

False, neither standard vector is in the subspace {[u₁ u₂] in the set R^2 : u₁+u₂ = 0}

The row space of an m ×n matrix A is the set {Av: v is in R^n}

False, the column space of an m x n matrix equal {Av:v is in R^n}

The determinant of a square matrix equals the product of its diagonal entries

False

The column space of any matrix equals the column space of its reduced row echelon form

False A = {{1 ,2 },{ 1, 2}} the rref is {{1 ,2 },{ 0, 0}}

If (t −4)² divides the characteristic polynomial of A, then 4 is an eigenvalue of A with multiplicity 2

False, consider 4*I₃; here 4 is an eigenvalue of multiplicity 3

To rotate the coordinate axes in order to remove the xy-term of the equation ax2 +2bxy +cy2 =d, we must determine the eigenvectors of {{a,b},{c,d}}

False, the correct matrix {{a,b},{b,c}}

The determinant of any square matrix equals the product of the diagonal entries of its reduced row echelon form.

False, the determinant of 2*I₂ is 4, but its reduced row echelon form is Identity 2.

The determinant of an upper triangular n ×n matrix or a lower triangular n ×n matrix equals the sum of its diagonal entries

False, the determinant of an upper triangular or a lower triangular square matrix equals the product of its diagonal entries

The dimension of the null space of a matrix equals the rank of the matrix

False, the dimension of the null space of a matrix equals the nullity of the matrix

If u (dot) v=0, then u=0 or v=0.

False, consider nonzero orthogonal vectors.

Combining the vectors in two orthonormal subsets of R^n produces another orthonormal subset of R^n

False, consider the sets {e₁} and {-e₁}. The combined set is {e₁,-e₁}, which are not orthonormal.

If x is orthogonal to y and y is orthogonal to z, then x is orthogonal to z

False, consider x = e₁, y = 0, and z = e₁

The norm of a sum of vectors is the sum of the norms of the vectors

False, for example, if v is a nonzero vector, then ||v+(-v)|| = 0 ≠ ||v||+||-v||