Subspaces in $\R^n$

Subspaces in $\R^n$

Definition

Let $A$ be an $m \times n$ real matrix.

  1. A subset $W$ in $\R^n$ is called a subspace if $W$ is a vector space in $\R^n$.
  2. The null space $\calN(A)$ of $A$ is defined by
    \[ \calN(A)=\{ \mathbf{x}\in \R^n \mid A\mathbf{x}=\mathbf{0}_m\}.\]
  3. The range $\calR(A)$ of the matrix $A$ is
    \[ \calR(A)=\{\mathbf{y} \in \R^m \mid \mathbf{y}=A\mathbf{x} \text{ for some } \mathbf{x} \in \R^n\}.\]
  4. The column space of $A$ is the subspace of $A^m$ spanned by the columns vectors of $A$.
  5. The row space of $A$ is the subspace of $A^n$ spanned by the rows vectors of $A$.
Summary

Let $A$ be an $m \times n$ real matrix.

  1. (Subspace Criteria) A subset $W$ in $\R^n$ is a subspace of $\R^n$ if and only if the following three condisions are met.
    (a) The zero vector $\mathbf{0} \in \R^n$ is in $W$.
    (b) If $\mathbf{x}, \mathbf{y} \in W$, then $\mathbf{x}+\mathbf{y}\in W$.
    (c) If $\mathbf{x} \in W$ and $c\in \R$, then $c\mathbf{x} \in W$.
  2. The nullspace of $A$ is a subspace in $\R^n$.
  3. The range of $A$ is a subspace in $\R^m$.
  4. The range of $A$ is the column space of $A$.
  5. If the columns of $A$ are linearly independent, then $\calN(A)=\{\mathbf{0}\}$.

=solution

Problems

  1. Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.
    (1) \[S_1=\left \{\, \begin{bmatrix}
    x_1 \\
    x_2 \\
    x_3
    \end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\] in the vector space $\R^3$.
    (2) \[S_2=\left \{\, \begin{bmatrix}
    x_1 \\
    x_2 \\
    x_3
    \end{bmatrix} \in \R^3 \quad \middle | \quad x_1-4x_2+5x_3=2 \,\right \}\] in the vector space $\R^3$.
    (3) \[S_3=\left \{\, \begin{bmatrix}
    x \\
    y
    \end{bmatrix}\in \R^2 \quad \middle | \quad y=x^2 \quad \,\right \}\] in the vector space $\R^2$.
    (4) Let $P_4$ be the vector space of all polynomials of degree $4$ or less with real coefficients.
    \[S_4=\{ f(x)\in P_4 \mid f(1) \text{ is an integer}\}\] in the vector space $P_4$.
    (5) \[S_5=\{ f(x)\in P_4 \mid f(1) \text{ is a rational number}\}\] in the vector space $P_4$.
    (6) Let $M_{2 \times 2}$ be the vector space of all $2\times 2$ real matrices.
    \[S_6=\{ A\in M_{2\times 2} \mid \det(A) \neq 0\} \] in the vector space $M_{2\times 2}$.
    (7) \[S_7=\{ A\in M_{2\times 2} \mid \det(A)=0\} \] in the vector space $M_{2\times 2}$.
    (8) Let $C[-1, 1]$ be the vector space of all real continuous functions defined on the interval $[a, b]$.
    \[S_8=\{ f(x)\in C[-2,2] \mid f(-1)f(1)=0\} \] in the vector space $C[-2, 2]$.
    (9) \[S_9=\{ f(x) \in C[-1, 1] \mid f(x)\geq 0 \text{ for all } -1\leq x \leq 1\}\] in the vector space $C[-1, 1]$.
    (10) Let $C^2[a, b]$ be the vector space of all real-valued functions $f(x)$ defined on $[a, b]$, where $f(x), f'(x)$, and $f^{\prime\prime}(x)$ are continuous on $[a, b]$. Here $f'(x), f^{\prime\prime}(x)$ are the first and second derivative of $f(x)$.
    \[S_{10}=\{ f(x) \in C^2[-1, 1] \mid f^{\prime\prime}(x)+f(x)=\sin(x) \text{ for all } -1\leq x \leq 1\}\] in the vector space $C[-1, 1]$.

  2. An $n\times n$ matrix $A$ is called orthogonal if $A^{\trans}A=I$.
    Let $V$ be the vector space of all real $2\times 2$ matrices. Consider the subset
    \[W:=\{A\in V \mid \text{$A$ is an orthogonal matrix}\}.\] Prove or disprove that $W$ is a subspace of $V$.

  3. Let $A$ be an $m \times n$ matrix. Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$. Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$. Then find $A\mathbf{w}$.

  4. Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by
    \[W=\left\{ \mathbf{x}=\begin{bmatrix}
    x_1 \\
    x_2 \\
    x_3
    \end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.\] (a) Which of the following vectors are in the subset $W$? Choose all vectors that belong to $W$.
    \[(1) \begin{bmatrix}
    0 \\
    0 \\
    0
    \end{bmatrix} \qquad(2) \begin{bmatrix}
    1 \\
    2 \\
    2
    \end{bmatrix} \qquad(3)\begin{bmatrix}
    3 \\
    0 \\
    0
    \end{bmatrix} \qquad(4) \begin{bmatrix}
    0 \\
    0
    \end{bmatrix} \qquad(5) \begin{bmatrix}
    1 & 2 & 4 \\
    1 &2 &4
    \end{bmatrix} \qquad(6) \begin{bmatrix}
    1 \\
    -1 \\
    -2
    \end{bmatrix}.\] (b) Determine whether $W$ is a subspace of $\R^3$ or not.

  5. Let $A=\begin{bmatrix}
    1 & 2 & 1 \\
    3 &6 &4
    \end{bmatrix}$ and let
    \[\mathbf{a}=\begin{bmatrix}
    -3 \\
    1 \\
    1
    \end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix}
    -2 \\
    1 \\
    0
    \end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix}
    1 \\
    1
    \end{bmatrix}.\] For each of the vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$, determine whether the vector is in the null space $\calN(A)$. Do the same for the range $\calR(A)$.

  6. Let $W$ be the subset of $\R^3$ defined by
    \[W=\left\{ \mathbf{x}=\begin{bmatrix}
    x_1 \\
    x_2 \\
    x_3
    \end{bmatrix} \in \R^3 \quad \middle| \quad x_1=3x_2 \text{ and } x_3=0 \right\}.\] Determine whether the subset $W$ is a subspace of $\R^3$ or not.

  7. Let
    \[A=\begin{bmatrix}
    1 & 2 & 2 \\
    2 &3 &2 \\
    -1 & -3 & -4
    \end{bmatrix} \text{ and }
    B=\begin{bmatrix}
    1 & 2 & 2 \\
    2 &3 &2 \\
    5 & 3 & 3
    \end{bmatrix}.\] Determine the null spaces of matrices $A$ and $B$.

  8. Let $W$ be the subset of $\R^3$ defined by
    \[W=\left \{ \mathbf{x}=\begin{bmatrix}
    x_1 \\
    x_2 \\
    x_3
    \end{bmatrix}\in \R^3 \quad \middle| \quad 5x_1-2x_2+x_3=0 \right \}.\] Exhibit a $1\times 3$ matrix $A$ such that $W=\calN(A)$, the null space of $A$. Conclude that the subset $W$ is a subspace of $\R^3$.

  9. Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$. Then prove that $V$ is a subspace of $\R^n$.

  10. Prove that the null space $\calN(A)$ is a subspace of the vector space $\R^n$.

  11. Prove that every plane in the $3$-dimensional space $\R^3$ that passes through the origin is a subspace of $\R^3$.
  12. Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}
    x \\
    y \\
    z \\
    w
    \end{bmatrix}$ satisfying $2x+3y+5z+7w=0$. Then prove that the set $S$ is a subspace of $\R^4$.
    (The Ohio State University)

  13. (a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}
    x \\
    y \\
    z \\
    w
    \end{bmatrix}$ satisfying
    \[2x+4y+3z+7w+1=0.\] Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.
    (b) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}
    x \\
    y \\
    z \\
    w
    \end{bmatrix}$ satisfying
    \[2x+4y+3z+7w=0.\] Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.
    (The Ohio State University)

  14. Let $S$ be the following subset of the 3-dimensional vector space $\R^3$.
    \[S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix}
    x_1 \\
    x_2 \\
    x_3
    \end{bmatrix}, x_1, x_2, x_3 \in \Z \right\}, \] where $\Z$ is the set of all integers. Determine whether $S$ is a subspace of $\R^3$.

  15. Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by
    \[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.\] Prove that the subset $W$ is a subspace of $\R^3$.

  16. Let $U$ and $V$ be subspaces of the $n$-dimensional vector space $\R^n$. Prove that the intersection $U\cap V$ is also a subspace of $\R^n$.
  17. Determine whether the following is true or false. If it is true, then give a proof. If it is false, then give a counterexample. Let $W_1$ and $W_2$ be subspaces of the vector space $\R^n$.If $B_1$ and $B_2$ are bases for $W_1$ and $W_2$, respectively, then $B_1\cap B_2$ is a basis of the subspace $W_1\cap W_2$.

  18. Let $U$ and $V$ be subspaces of the vector space $\R^n$. If neither $U$ nor $V$ is a subset of the other, then prove that the union $U \cup V$ is not a subspace of $\R^n$.

  19. Let $W_1, W_2$ be subspaces of a vector space $V$. Then prove that $W_1 \cup W_2$ is a subspace of $V$ if and only if $W_1 \subset W_2$ or $W_2 \subset W_1$.
  20. Let $A$ and $B$ be $n\times n$ matrices. Then prove that
    \[\calN(A)\cap \calN(B) \subset \calN(A+B).\]