TRUE: The rows become the columns of AT so this makes sense.

The row space of A is the same as the column space of A^T

FALSE: The nonzero rows of B form a basis. The first three rows of A may be linear dependent

If B is an echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis of Row A.

TRUE by the Rank Theorem. Also since dimension of row space = number of nonzero rows in echelon form = number pivot columns = dimension of column space

The dimensions of the row space and the column space of A are the same, even if A if A is not square.

FALSE: Equals number of columns by rank theorem. Also dimension of row space = number pivot columns, dimension of null space = number of non-pivot columns (free variables) so these add to total number of columns

The sum of the dimensions of the row space and the null space of A equals the number of rows in A.

TRUE: Due to rounding error.

On a computer, row operations can change the apparent rank of a matrix.

FALSE: It’s the corresponding columns in A

If B is any echelon form of A, the the pivot columns of B form a basis for the column space of A.

FALSE: For example, Row interchanges mess things up.

Row operations preserve the linear dependence relations among the rows of A .

TRUE: These correspond with the free variables

The dimension of null space of A is the number of columns of A that are not pivot columns.

TRUE Columns of A go to rows of A^T

The row space of A^T is the same as the column space of A.

TRUE: If spanned by three vectors must be all of R3

The only three dimensional subspace of R3 is R3 itself.

FALSE: S must have exactly n elements or be noted as linearly independent

If dim(V)=n and if S spans V then S is a basis for V

FALSE: It must be impossible to span it by a finite

set.

set.

A vector space is infinite dimensional is it is spanned by an infinite set.

FALSE: Not a subset

R2 is a two dimensional subspace of R3.

TRUE: The number of linearly independent

vectors that span a set is unique

vectors that span a set is unique

If a set {v1:::vn} spans a finite dimensional vector space

V and if T is a set of more than n vectors in V, then T is linearly dependent.

V and if T is a set of more than n vectors in V, then T is linearly dependent.

FALSE: S must have exactly n elements

If dim(V)=n and S is a linearly independent set in V, then S is a basis for V.

FALSE: It’s 5.

The dimension of the vector space P4 is 4.

FALSE: unless the plane is through the origin

A plane in R3 is a two dimensional subspace of R3.

TRUE: Remember these columns and linearly independent and span the column space

The number of pivot columns of a matrix equals the dimension of its column space.

FALSE: Must look at

corresponding columns in A

corresponding columns in A

If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.

FALSE: It NEVER fails!!!

The standard method for producing a spanning set for Nul(A), described in this section, sometimes fails to produce a basis.

TRUE

A basis is a linearly independent set that is as large as possible.

TRUE: by Spanning Set Theorem

If a finite set S of nonzero vectors spans a vector space V, the some subset is a basis for V.

FALSE: It may not span

A linearly independent set in a subspace H is a basis for H.

FALSE: They are not affected

In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix.

FALSE: It is is too large, then it is no longer linearly independent

A basis is a spanning set that is as large as possible.

TRUE: They are linearly independent and span Rn.

The columns of an invertible n x n matrix form a basis for Rn

FALSE: They may not be linearly independent

If H =Span {b1;:::;bn} then {b1;:::;bn} is a basis for H.

FALSE: unless it in the zero vector

A single vector is itself linearly dependent.

TRUE

The set of all solutions of a homogenous linear differential equation is the kernel of a linear transformation.

TRUE: It’s a subspace(check), thus vector space.

The range of a linear transformation is a vector space.

TRUE

Nul(A) is the kernel of the mapping x -> Ax.

FALSE: It is the set of all b that have solutions

Col(A) is the set of all solutions of Ax=b.

TRUE

The column space of an m x n matrix is in Rm

TRUE

The null space is a vector space.

TRUE Remember that Ax gives a linear combination of columns of A using x entries as weights.

Col(A) is the set of a vectors that can be written as Ax for some x.

TRUE: To show this we show it is a subspace

The kernel of a linear transformation is a vector space.

FALSE: must be consistent for all b

If the equation Ax = b is consistent, then Col(A) is Rm.

TRUE

The column space of A is the range of the mapping

x -> Ax.

x -> Ax.

False. It’s Rn

The null space of an m x n matrix is in Rm.

TRUE

The null space of A is the solution set of the equation

Ax = 0.

Ax = 0.

TRUE (Its always a subspace of itself, at the very least.)

A vector space is also a subspace.