LINEAR ALGEBRA EXAM #2
QUESTIONS AND ANSWERS
row replacement operation does not affect the determinant of a matrix. - Answer-True. If
a multiple of one row of a matrix A is added to another to produce a matrix B, then det B
equals det A
The determinant of A is the product of the pivots in any echelon form U of A, multiplied
by (-1)^r, where r is the number of row interchanges made during row reduction from A
to U. - Answer-False. Reduction to an echelon form may also include scaling a row by a
nonzero constant, which can change the value of the determinant.
If the columns of A are linearly dependent, then det A=0. - Answer-True. If the columns
of A are linearly dependent, then A is not invertible
det(A+B)=det A+det B - Answer-False
The set of all polynomials of the form p(t)=at^4, where a is in set of real numbers ℝ. -
Answer-The set is a subspace of set of ℙ4. The set contains the zero vector of set of ℙ4
, the set is closed under vector addition, and the set is closed under multiplication by
scalars.
The set of all polynomials in set of ℙn such that p(0)=0 - Answer-.
The set is a subspace of set of ℙn because the set contains the zero vector of set of ℙn,
the set is closed under vector addition, and the set is closed under multiplication by
scalars
A null space is a vector space. - Answer-True because the null space of an m×n matrix
A is a subspace ℝn
The column space of an m×n matrix is in ℝm. - Answer-True because the column space
of an m×n matrix A is a subspace of ℝm.
The column space of A, Col(A), is the set of all solutions of Ax=b. - Answer-False
because Col(A)={b : b=Ax for some x in ℝn}
The null space of A, Nul(A), is the kernel of the mapping x-->Ax - Answer-True, the
kernel of a linear transformation T, from a vector space V to a vector space W, is the set
of all u in V such that T(u)=0. Thus, the kernel of a matrix transformation T(x)=Ax is the
null space of A.
The range of a linear transformation is a vector space. - Answer-True, the range of a
linear transformation T, from a vector space V to a vector space W, is a subspace of W.
QUESTIONS AND ANSWERS
row replacement operation does not affect the determinant of a matrix. - Answer-True. If
a multiple of one row of a matrix A is added to another to produce a matrix B, then det B
equals det A
The determinant of A is the product of the pivots in any echelon form U of A, multiplied
by (-1)^r, where r is the number of row interchanges made during row reduction from A
to U. - Answer-False. Reduction to an echelon form may also include scaling a row by a
nonzero constant, which can change the value of the determinant.
If the columns of A are linearly dependent, then det A=0. - Answer-True. If the columns
of A are linearly dependent, then A is not invertible
det(A+B)=det A+det B - Answer-False
The set of all polynomials of the form p(t)=at^4, where a is in set of real numbers ℝ. -
Answer-The set is a subspace of set of ℙ4. The set contains the zero vector of set of ℙ4
, the set is closed under vector addition, and the set is closed under multiplication by
scalars.
The set of all polynomials in set of ℙn such that p(0)=0 - Answer-.
The set is a subspace of set of ℙn because the set contains the zero vector of set of ℙn,
the set is closed under vector addition, and the set is closed under multiplication by
scalars
A null space is a vector space. - Answer-True because the null space of an m×n matrix
A is a subspace ℝn
The column space of an m×n matrix is in ℝm. - Answer-True because the column space
of an m×n matrix A is a subspace of ℝm.
The column space of A, Col(A), is the set of all solutions of Ax=b. - Answer-False
because Col(A)={b : b=Ax for some x in ℝn}
The null space of A, Nul(A), is the kernel of the mapping x-->Ax - Answer-True, the
kernel of a linear transformation T, from a vector space V to a vector space W, is the set
of all u in V such that T(u)=0. Thus, the kernel of a matrix transformation T(x)=Ax is the
null space of A.
The range of a linear transformation is a vector space. - Answer-True, the range of a
linear transformation T, from a vector space V to a vector space W, is a subspace of W.
