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Tagged: leading 1 method
by Yu · Published 02/26/2018
Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors
Problem 710
Find a basis for $\Span(S)$ where $S=
\left\{
\begin{bmatrix}
1 \\ 2 \\ 1
\end{bmatrix}
-1 \\ -2 \\ -1
2 \\ 6 \\ -2
\right\}$.
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Find a Basis for the Subspace spanned by Five Vectors
Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where
\[
\mathbf{v}_{1}=
1 \\ 2 \\ 2 \\ -1
,\;\mathbf{v}_{2}=
1 \\ 3 \\ 1 \\ 1
1 \\ 5 \\ -1 \\ 5
.\] Find a basis for the span $\Span(S)$.
Click here if solved 111
Find a Basis for Nullspace, Row Space, and Range of a Matrix
Let $A=\begin{bmatrix}
2 & 4 & 6 & 8 \\
1 &3 & 0 & 5 \\
1 & 1 & 6 & 3
\end{bmatrix}$.
(a) Find a basis for the nullspace of $A$.
(b) Find a basis for the row space of $A$.
(c) Find a basis for the range of $A$ that consists of column vectors of $A$.
(d) For each column vector which is not a basis vector that you obtained in part (c), express it as a linear combination of the basis vectors for the range of $A$.
by Yu · Published 11/08/2017 · Last modified 11/10/2017
Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less
Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less.
\[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\] where
\begin{align*}
p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\
p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3.
\end{align*}
(a) Find a basis $Q$ of the span $\Span(S)$ consisting of polynomials in $S$.
(b) For each polynomial in $S$ that is not in $Q$, find the coordinate vector with respect to the basis $Q$.
(The Ohio State University, Linear Algebra Midterm)
Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors
Let $V$ be a vector space and $B$ be a basis for $V$.
Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$.
Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ with respect to the basis $B$.
After applying the elementary row operations to $A$, we obtain the following matrix in reduced row echelon form
\[\begin{bmatrix}
1 & 0 & 2 & 1 & 0 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix}.\]
(a) What is the dimension of $V$?
(b) What is the dimension of $\Span\{\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5\}$?
Find Bases for the Null Space, Range, and the Row Space of a $5\times 4$ Matrix
\[A=\begin{bmatrix}
1 & -1 & 0 & 0 \\
0 & 2 & 2 & 2\\
(a) Find a basis for the null space $\calN(A)$.
(b) Find a basis of the range $\calR(A)$.
(c) Find a basis of the row space for $A$.
Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials
Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less.
Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where
p_1(x)&=-1+x+2x^2, \quad p_2(x)=x+3x^2\\
p_3(x)&=1+2x+8x^2, \quad p_4(x)=1+x+x^2.
(a) Find a basis of $P_2$ among the vectors of $S$. (Explain why it is a basis of $P_2$.)
(b) Let $B'$ be the basis you obtained in part (a).
For each vector of $S$ which is not in $B'$, find the coordinate vector of it with respect to the basis $B'$.
(The Ohio State University, Linear Algebra Final Exam Problem)
Find an Orthonormal Basis of the Range of a Linear Transformation
Let $T:\R^2 \to \R^3$ be a linear transformation given by
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} \,\right)
x_1-x_2 \\
x_1+ x_2
\end{bmatrix}.\] Find an orthonormal basis of the range of $T$.
Rank and Nullity of Linear Transformation From $\R^3$ to $\R^2$
Let $T:\R^3 \to \R^2$ be a linear transformation such that
\[ T(\mathbf{e}_1)=\begin{bmatrix}
1 \\
\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}
\end{bmatrix},\] where $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ are the standard basis of $\R^3$.
Then find the rank and the nullity of $T$.
(The Ohio State University, Linear Algebra Exam Problem)
Orthonormal Basis of Null Space and Row Space
1 & 0 & 1 \\
0 &1 &0
(a) Find an orthonormal basis of the null space of $A$.
(b) Find the rank of $A$.
(c) Find an orthonormal basis of the row space of $A$.
Coordinate Vectors and Dimension of Subspaces (Span)
Let $V$ be a vector space over $\R$ and let $B$ be a basis of $V$.
Let $S=\{v_1, v_2, v_3\}$ be a set of vectors in $V$. If the coordinate vectors of these vectors with respect to the basis $B$ is given as follows, then find the dimension of $V$ and the dimension of the span of $S$.
\[[v_1]_B=\begin{bmatrix}
\end{bmatrix}, [v_2]_B=\begin{bmatrix}
Quiz 9. Find a Basis of the Subspace Spanned by Four Matrices
Let $V$ be the vector space of all $2\times 2$ real matrices.
Let $S=\{A_1, A_2, A_3, A_4\}$, where
\[A_1=\begin{bmatrix}
1 & 2\\
-1& 3
\end{bmatrix}, A_2=\begin{bmatrix}
0 & -1\\
1& 4
-1 & 0\\
1& -10
\end{bmatrix}.\] Then find a basis for the span $\Span(S)$.
Quiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix
(a) Let $A=\begin{bmatrix}
Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$.
(b) Find the rank and nullity of the matrix $A$ in part (a).
Given a Spanning Set of the Null Space of a Matrix, Find the Rank
Let $A$ be a real $7\times 3$ matrix such that its null space is spanned by the vectors
\end{bmatrix}, \begin{bmatrix}
\end{bmatrix}, \text{ and } \begin{bmatrix}
-1 \\
\end{bmatrix}.\] Then find the rank of the matrix $A$.
(Purdue University, Linear Algebra Final Exam Problem)
Linear Transformation, Basis For the Range, Rank, and Nullity, Not Injective
Let $V$ be the vector space of all $2\times 2$ real matrices and let $P_3$ be the vector space of all polynomials of degree $3$ or less with real coefficients.
Let $T: P_3 \to V$ be the linear transformation defined by
\[T(a_0+a_1x+a_2x^2+a_3x^3)=\begin{bmatrix}
a_0+a_2 & -a_0+a_3\\
a_1-a_2 & -a_1-a_3
\end{bmatrix}\] for any polynomial $a_0+a_1x+a_2x^2+a_3 \in P_3$.
Find a basis for the range of $T$, $\calR(T)$, and determine the rank of $T$, $\rk(T)$, and the nullity of $T$, $\nullity(T)$.
Also, prove that $T$ is not injective.
Row Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a Matrix
Let \[A=\begin{bmatrix}
2 &2 &4 \\
2 & 3 & 5
(a) Find a matrix $B$ in reduced row echelon form such that $B$ is row equivalent to the matrix $A$.
(b) Find a basis for the null space of $A$.
(c) Find a basis for the range of $A$ that consists of columns of $A$. For each columns, $A_j$ of $A$ that does not appear in the basis, express $A_j$ as a linear combination of the basis vectors.
(d) Exhibit a basis for the row space of $A$.
Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$
Define the map $T:\R^2 \to \R^3$ by $T \left ( \begin{bmatrix}
\end{bmatrix}\right )=\begin{bmatrix}
x_1+x_2 \\
(a) Show that $T$ is a linear transformation.
(b) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for each $\mathbf{x} \in \R^2$.
(c) Describe the null space (kernel) and the range of $T$ and give the rank and the nullity of $T$.
Find a Basis for a Subspace of the Vector Space of $2\times 2$ Matrices
Let $V$ be the vector space of all $2\times 2$ matrices, and let the subset $S$ of $V$ be defined by $S=\{A_1, A_2, A_3, A_4\}$, where
A_1=\begin{bmatrix}
1 & 2 \\
-1 & 3
\end{bmatrix}, \quad
0 & -1 \\
-1 & 0 \\
1 & -10
\end{bmatrix}.
Find a basis of the span $\Span(S)$ consisting of vectors in $S$ and find the dimension of $\Span(S)$.
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