Bases and Dimension of Subspaces in $\R^n$

Bases and Dimension of Subspaces in $\R^n$

Definition

Let $V$ be a subspace in $\R^n$.

  1. A basis for $V$ is a linearly independent spanning set for $V$.
  2. The number of vectors in a basis for $V$ is called the dimension of $V$.
  3. The dimension of the range $\calR(A)$ of a matrix $A$ is called the rank of $A$.
  4. The dimension of the null space $\calN(A)$ of a matrix $A$ is called the nullity of $A$.
Summary

  1. A basis is not unique.
  2. The rank-nullity theorem:
    (Rank of $A$)+(Nullity of $A$)=(The number of columns in $A$).
  3. The rank of $A$ and the rank of $A^{\trans}$ are the same.

=solution

Problems

  1. Let
    \[\mathbf{v}=\begin{bmatrix}
    a \\
    b \\
    c
    \end{bmatrix}, \qquad \mathbf{v}_1=\begin{bmatrix}
    1 \\
    2 \\
    0
    \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix}
    2 \\
    -1 \\
    2
    \end{bmatrix}.\] Find the necessary and sufficient condition so that the vector $\mathbf{v}$ is a linear combination of the vectors $\mathbf{v}_1, \mathbf{v}_2$.

  2. Find a basis of the null space of the matrix $B=\begin{bmatrix}
    1 & 1 & 2 \\
    -2 &-2 &-4
    \end{bmatrix}$.

  3. (a) Let $A=\begin{bmatrix}
    1 & 3 & 0 & 0 \\
    1 &3 & 1 & 2 \\
    1 & 3 & 1 & 2
    \end{bmatrix}$. 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).

  4. Let
    \[A=\begin{bmatrix}
    1 & 1 & 0 \\
    1 &1 &0
    \end{bmatrix}\] be a matrix. Find a basis of the null space of the matrix $A$.

  5. Let \[A=\begin{bmatrix}
    1 & 1 & 2 \\
    2 &2 &4 \\
    2 & 3 & 5
    \end{bmatrix}.\] (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 $A$.

  6. Let
    \[A=\begin{bmatrix}
    1 & -1 & 0 & 0 \\
    0 &1 & 1 & 1 \\
    1 & -1 & 0 & 0 \\
    0 & 2 & 2 & 2\\
    0 & 0 & 0 & 0
    \end{bmatrix}.\] (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 $A$.
    (The Ohio State University)

  7. Determine whether each of the following sets is a basis for $\R^3$.
    (a) $S=\left\{\, \begin{bmatrix}
    1 \\
    0 \\
    -1
    \end{bmatrix}, \begin{bmatrix}
    2 \\
    1 \\
    -1
    \end{bmatrix}, \begin{bmatrix}
    -2 \\
    1 \\
    4
    \end{bmatrix} \,\right\}$
    (b) $S=\left\{\, \begin{bmatrix}
    1 \\
    4 \\
    7
    \end{bmatrix}, \begin{bmatrix}
    2 \\
    5 \\
    8
    \end{bmatrix}, \begin{bmatrix}
    3 \\
    6 \\
    9
    \end{bmatrix} \,\right\}$
    (c) $S=\left\{\, \begin{bmatrix}
    1 \\
    1 \\
    2
    \end{bmatrix}, \begin{bmatrix}
    0 \\
    1 \\
    7
    \end{bmatrix} \,\right\}$
    (d) $S=\left\{\, \begin{bmatrix}
    1 \\
    2 \\
    5
    \end{bmatrix}, \begin{bmatrix}
    7 \\
    4 \\
    0
    \end{bmatrix}, \begin{bmatrix}
    3 \\
    8 \\
    6
    \end{bmatrix}, \begin{bmatrix}
    -1 \\
    9 \\
    10
    \end{bmatrix} \,\right\}$

  8. Suppose that the vectors
    \[\mathbf{v}_1=\begin{bmatrix}
    -2 \\
    1 \\
    0 \\
    0 \\
    0
    \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix}
    -4 \\
    0 \\
    -3 \\
    -2 \\
    1
    \end{bmatrix}\] are a basis vectors for the null space of a $4\times 5$ matrix $A$. Find a vector $\mathbf{x}$ such that $\mathbf{x}\neq 0$, $\mathbf{x}\neq \mathbf{v}_1$, $\mathbf{x}\neq \mathbf{v}_2$ and $A\mathbf{x}=\mathbf{0}$.
    (Stanford University)

  9. Let $V$ be the following subspace of the $4$-dimensional vector space $\R^4$.
    \[V:=\left\{ \quad\begin{bmatrix}
    x_1 \\
    x_2 \\
    x_3 \\
    x_4
    \end{bmatrix} \in \R^4
    \quad \middle| \quad
    x_1-x_2+x_3-x_4=0 \quad\right\}.\] Find a basis of the subspace $V$ and its dimension.

  10. A hyperplane in $n$-dimensional vector space $\R^n$ is defined to be the set of vectors $\begin{bmatrix}
    x_1 \\
    x_2 \\
    \vdots \\
    x_n
    \end{bmatrix}\in \R^n$ satisfying the linear equation of the form
    \[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\] where $a_1, a_2, \dots, a_n$ (at least one of $a_1, a_2, \dots, a_n$ is nonzero) and $b$ are real numbers. Here at least one of $a_1, a_2, \dots, a_n$ is nonzero. Consider the hyperplane $P$ in $\R^n$ described by the linear equation
    \[a_1x_1+a_2x_2+\cdots+a_nx_n=0,\] where $a_1, a_2, \dots, a_n$ are some fixed real numbers and not all of these are zero.
    (The constant term $b$ is zero.) Then prove that the hyperplane $P$ is a subspace of $R^{n}$ of dimension $n-1$.

  11. Let $A$ be a real $7\times 3$ matrix such that its null space is spanned by the vectors
    \[\begin{bmatrix}
    1 \\
    2 \\
    0
    \end{bmatrix}, \begin{bmatrix}
    2 \\
    1 \\
    0
    \end{bmatrix}, \text{ and } \begin{bmatrix}
    1 \\
    -1 \\
    0
    \end{bmatrix}.\] Then find the rank of the matrix $A$.
    (Purdue University)

  12. Let $A=\begin{bmatrix}
    4 & 1\\
    3& 2
    \end{bmatrix}$ and consider the following subset $V$ of the 2-dimensional vector space $\R^2$.
    \[V=\{\mathbf{x}\in \R^2 \mid A\mathbf{x}=5\mathbf{x}\}.\] (a) Prove that the subset $V$ is a subspace of $\R^2$.
    (b) Find a basis for $V$ and determine the dimension of $V$.

  13. Let $V$ be a subset of $\R^4$ consisting of vectors that are perpendicular to vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, where
    \[\mathbf{a}=\begin{bmatrix}
    1 \\
    0 \\
    1 \\
    0
    \end{bmatrix}, \quad \mathbf{b}=\begin{bmatrix}
    1 \\
    1 \\
    0 \\
    0
    \end{bmatrix}, \quad \mathbf{c}=\begin{bmatrix}
    0 \\
    1 \\
    -1 \\
    0
    \end{bmatrix}.\] Namely,
    \[V=\{\mathbf{x}\in \R^4 \mid \mathbf{a}^{\trans}\mathbf{x}=0, \mathbf{b}^{\trans}\mathbf{x}=0, \text{ and } \mathbf{c}^{\trans}\mathbf{x}=0\}.\] (a) Prove that $V$ is a subspace of $\R^4$.
    (b) Find a basis of $V$.
    (c) Determine the dimension of $V$.

  14. Find a basis for the subspace $W$ of all vectors in $\R^4$ which are perpendicular to the columns of the matrix
    \[A=\begin{bmatrix}
    11 & 12 & 13 & 14 \\
    21 &22 & 23 & 24 \\
    31 & 32 & 33 & 34 \\
    41 & 42 & 43 & 44
    \end{bmatrix}.\] (Harvard University Exam)

  15. Let $V$ be the vector space over $\R$ of all real $2\times 2$ matrices. Let $W$ be the subset of $V$ consisting of all symmetric matrices.
    (a) Prove that $W$ is a subspace of $V$.
    (b) Find a basis of $W$.
    (c) Determine the dimension of $W$.

  16. Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r$ are linearly dependent $n$-dimensional real vectors. For any vector $\mathbf{v}_{r+1} \in \R^n$, determine whether the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_r, \mathbf{v}_{r+1}$ are linearly independent or linearly dependent.

  17. Let $B=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of three-dimensional vectors in $\R^3$.
    (a) Prove that if the set $B$ is linearly independent, then $B$ is a basis of the vector space $\R^3$.
    (b) Prove that if the set $B$ spans $\R^3$, then $B$ is a basis of $\R^3$.

  18. Let $V$ be a subspace of $\R^n$. Suppose that $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}$ is a spanning set for $V$. Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.
  19. Let $V$ be a subspace of $\R^n$. Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ is a basis of the subspace $V$. Prove that every basis of $V$ consists of $k$ vectors in $V$.
  20. Prove the followings.
    (a) $\calN(A)=\calN(A^{\trans}A)$.
    (b) $\rk(A)=\rk(A^{\trans}A)$.

  21. Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings.
    (a) $\rk(AB) \leq \rk(A)$.
    (b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$.

  22. Let $V$ and $W$ be subspaces of $\R^n$ such that $V \cap W =\{\mathbf{0}\}$ and $\dim(V)+\dim(W)=n$.
    (a) If $\mathbf{v}+\mathbf{w}=\mathbf{0}$, where $\mathbf{v}\in V$ and $\mathbf{w}\in W$, then show that $\mathbf{v}=\mathbf{0}$ and $\mathbf{w}=\mathbf{0}$.
    (b) If $B_1$ is a basis for the subspace $V$ and $B_2$ is a basis for the subspace $W$, then show that the union $B_1\cup B_2$ is a basis for $R^n$.
    (c) If $\mathbf{x}$ is in $\R^n$, then show that $\mathbf{x}$ can be written in the form $\mathbf{x}=\mathbf{v}+\mathbf{w}$, where $\mathbf{v}\in V$ and $\mathbf{w} \in W$.
    (d) Show that the representation obtained in part (c) is unique.

  23. Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.