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Linear Combination and Linear Independence

Linear Combination and Linear Independence

Definition

  1. The expression $c_1\mathbf{v}_1+c_2\mathbf{v}_2+\cdots+c_k\mathbf{v}_k$ is called a linear combination of vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\in \R^n$, where $c_1, c_2, \dots, c_k$ are scalars in $\R$.
  2. A set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ is said to be linearly independent if the only scalrs $c_1, c_2, \dots, c_k$ satisfying $c_1\mathbf{v}_1+c_2\mathbf{v}_2+\cdots+c_k\mathbf{v}_k=\mathbf{0}$ are $c_1=c_2=\cdots=c_k=0$. We also say that the vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ are linearly independent.
  3. If vectors are not linearly independent, they are linearly dependent.
Summary

  1. The set of $n$-dimensional vectors $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ are linearly dependent if $k > n$. (If there are more vectors than the dimension, then the vectors are linearly dependent.)

=solution

Problems

  1. Express the vector $\mathbf{b}=\begin{bmatrix}
    2 \\
    13 \\
    6
    \end{bmatrix}$ as a linear combination of the vectors
    \[\mathbf{v}_1=\begin{bmatrix}
    1 \\
    5 \\
    -1
    \end{bmatrix},
    \mathbf{v}_2=
    \begin{bmatrix}
    1 \\
    2 \\
    1
    \end{bmatrix},
    \mathbf{v}_3=
    \begin{bmatrix}
    1 \\
    4 \\
    3
    \end{bmatrix}.\] (The Ohio State University, Linear Algebra Exam)

  2. Write the vector $\begin{bmatrix} 1 \\ 3 \\ -1 \end{bmatrix}$ as a linear combination of the vectors $\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ , $\begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix}$, $\begin{bmatrix} 2 \\ 0 \\ 4 \end{bmatrix}$.
  3. For what value(s) of $a$ is the following set $S$ linearly dependent?
    \[ S=\left \{\,\begin{bmatrix}
    1 \\
    2 \\
    3 \\
    a
    \end{bmatrix}, \begin{bmatrix}
    a \\
    0 \\
    -1 \\
    2
    \end{bmatrix}, \begin{bmatrix}
    0 \\
    0 \\
    a^2 \\
    7
    \end{bmatrix}, \begin{bmatrix}
    1 \\
    a \\
    1 \\
    1
    \end{bmatrix}, \begin{bmatrix}
    2 \\
    -2 \\
    3 \\
    a^3
    \end{bmatrix} \, \right\}.\]
    See (a)
  4. Prove that any set of vectors which contains the zero vector is linearly dependent.
  5. Let $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ be a set of nonzero vectors in $\R^m$ such that the dot product $mathbf{v}_i\cdot \mathbf{v}_j=0$ when $i\neq j$. Prove that the set is linearly independent.
    See (b)
  6. Determine whether the following set of vectors is linearly independent or linearly dependent. If the set is linearly dependent, express one vector in the set as a linear combination of the others.
    \[\left\{\, \begin{bmatrix}
    1 \\
    0 \\
    -1 \\
    0
    \end{bmatrix}, \begin{bmatrix}
    1 \\
    2 \\
    3 \\
    4
    \end{bmatrix}, \begin{bmatrix}
    -1 \\
    -2 \\
    0 \\
    1
    \end{bmatrix},
    \begin{bmatrix}
    -2 \\
    -2 \\
    7 \\
    11
    \end{bmatrix}\, \right\}.\]
  7. Find the value(s) of $h$ for which the following set of vectors
    \[\left \{ \mathbf{v}_1=\begin{bmatrix}
    1 \\
    0 \\
    0
    \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
    h \\
    1 \\
    -h
    \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
    1 \\
    2h \\
    3h+1
    \end{bmatrix}\right\}\] is linearly independent.
    (Boston College)

  8. Let
    \[\mathbf{v}_1=\begin{bmatrix}
    1 \\
    2 \\
    0
    \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
    1 \\
    a \\
    5
    \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
    0 \\
    4 \\
    b
    \end{bmatrix}\] be vectors in $\R^3$. Determine a condition on the scalars $a, b$ so that the set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly dependent.

  9. Determine conditions on the scalars $a, b$ so that the following set $S$ of vectors is linearly dependent.
    \begin{align*}
    S=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\},
    \end{align*}
    where
    \[\mathbf{v}_1=\begin{bmatrix}
    1 \\
    3 \\
    1
    \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}
    1 \\
    a \\
    4
    \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}
    0 \\
    2 \\
    b
    \end{bmatrix}.\]

  10. Let $A$ be a $3\times 3$ matrix and let $\mathbf{v}=\begin{bmatrix}
    1 \\
    2 \\
    -1
    \end{bmatrix}$ and $\mathbf{w}=\begin{bmatrix}
    2 \\
    -1 \\
    3
    \end{bmatrix}$. Suppose that $A\mathbf{v}=-\mathbf{v}$ and $A\mathbf{w}=2\mathbf{w}$.
    Then find the vector $A^5\begin{bmatrix}
    -1 \\
    8 \\
    -9
    \end{bmatrix}$.

  11. (a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly dependent.
    (b) Prove that the row vectors of every $5\times 3$ matrix $B$ are linearly dependent.

  12. Suppose $M$ is an $n \times n$ upper-triangular matrix. If the diagonal entries of $M$ are all non-zero, then prove that the column vectors are linearly independent. Does the conclusion hold if we do not assume that $M$ has non-zero diagonal entries?
  13. Suppose that an $n \times m$ matrix $M$ is composed of the column vectors $\mathbf{b}_1 , \cdots , \mathbf{b}_m$. Prove that a vector $\mathbf{v} \in \R^n$ can be written as a linear combination of the column vectors if and only if there is a vector $\mathbf{x}$ which solves the equation $M \mathbf{x} = \mathbf{v}$.