# Linear Algebra True/False

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.
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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.
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
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.
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
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.
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.
TRUE (Its always a subspace of itself, at the very least.)
A vector space is also a subspace.

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