Linear independence

We would like to develop a means to detect when a set of vectors is linearly dependent. This activity will point the way.

1. Suppose we have five vectors in $\real^4$ that form the columns of a matrix having reduced row echelon form . Is it possible to write one of the vectors $\vvec_1,\vvec_2,\ldots,\vvec_5$ as a linear combination of the others? If so, show explicitly how one vector appears as a linear combination of some of the other ve…

1. Explain why the homogeneous equation $A\xvec = \zerovec$ is consistent no matter the matrix $A$.

2. Consider the matrix whose columns we denote by $\vvec_1$, $\vvec_2$, and $\vvec_3$. Describe the solution space of the homogeneous equation $A\xvec = \zerovec$ using a parametric description, if appropriate.

3. Find a nonzero solution to the homogeneous equation and use it to find weights $c_1$, $c_2$, and $c_3$ such that .

4. Use the equation you found in the previous part to write…

Consider the set of vectors .

1. Explain why this set of vectors is linearly dependent.

2. Write one of the vectors as a linear combination of the others.

3. Find weights $c_1$, $c_2$, $c_3$, and $c_4$, not all of which are zero, such that .

4. Suppose $A=\left[\begin{array}{rrrr} \vvec_1&\vvec_2&\vvec_3&\vvec_4 \end{array}\right]$. Find a nonzero solution to the homogenous equation $A\xvec = \zerovec$.

Consider the vectors .

1. Are these vectors linearly independent or linearly dependent?

2. Describe the $\laspan{\vvec_1,\vvec_2,\vvec_3}$.

3. Suppose that $\bvec$ is a vector in $\real^3$. Explain why we can guarantee that $\bvec$ may be written as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$.

4. Suppose that $\bvec$ is a vector in $\real^3$. In how many ways can $\bvec$ be written as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$?

Respond to the following questions and provide a justification for your responses.

1. If the vectors $\vvec_1$ and $\vvec_2$ form a linearly dependent set, must one vector be a scalar multiple of the other?

2. Suppose that $\vvec_1,\vvec_2,\ldots,\vvec_n$ is a linearly independent set of vectors. What can you say about the linear independence or dependence of a subset of these vectors?

3. Suppose $\vvec_1,\vvec_2,\ldots,\vvec_n$ is a linearly independent set of vectors that form the co…

Determine whether the following statements are true or false and provide a justification for your response.

1. If $\vvec_1,\vvec_2,\ldots,\vvec_n$ are linearly dependent, then one vector is a scalar multiple of one of the others.

2. If $\vvec_1, \vvec_2, \ldots, \vvec_{10}$ are vectors in $\real^5$, then the set of vectors is linearly dependent.

3. If $\vvec_1, \vvec_2, \ldots, \vvec_{5}$ are vectors in $\real^{10}$, then the set of vectors is linearly independent.

4. Suppose we have a set of vectors $\vvec_1,\vvec_2,\ldots,\vvec_n$ and that $\vvec_2$ is a scalar multiple…

Suppose we have a set of vectors $\vvec_1,\vvec_2,\vvec_3,\vvec_4$ in $\real^5$ that satisfy the relationship: and suppose that $A$ is the matrix $A=\left[\begin{array}{rrrr} \vvec_1&\vvec_2&\vvec_3&\vvec_4 \end{array}\right]$.

1. Find a nonzero solution to the equation $A\xvec = \zerovec$.

2. Explain why the matrix $A$ has a column without a pivot position.

3. Write one of the vectors as a linear combination of the others.

4. Explain why the set of vectors is linearly dependent.

Suppose that $\vvec_1,\vvec_2,\ldots,\vvec_n$ is a set of vectors in $\real^{27}$ that form the columns of a matrix $A$.

1. Suppose that the vectors span $\real^{27}$. What can you say about the number of vectors $n$ in this set?

2. Suppose instead that the vectors are linearly independent. What can you say about the number of vectors $n$ in this set?

3. Suppose that the vectors are both linearly independent and span $\real^{27}$. What can you say about the number of vectors in…

Given below are some descriptions of sets of vectors that form the columns of a matrix $A$. For each description, give a possible reduced row echelon form for $A$ or indicate why there is no set of vectors satisfying the description by stating why the required reduced row echelon matrix cannot exist.

1. A set of 4 linearly independent vectors in $\real^5$.

2. A set of 4 linearly independent vectors in $\real^4$.

3. A set of 3 ve…

Suppose that $k$ is an unknown parameter and consider the set of vectors .

1. For what values of $k$ is the set of vectors linearly dependent?

2. For what values of $k$ does the set of vectors span $\real^3$?

Given a set of linearly dependent vectors, we can eliminate some of the vectors to create a smaller, linearly independent set of vectors.

1. Suppose that $\wvec$ is a linear combination of the vectors $\vvec_1$ and $\vvec_2$. Explain why $\laspan{\vvec_1,\vvec_2, \wvec} = \laspan{\vvec_1,\vvec_2}$.

2. Consider the vectors . Write one of the vectors as a linear combination of the others. Find a set of three vectors whose span is the same as $\laspan{\vvec_1,\vvec_2,\vvec_3,\vvec_4}$.

3. Are the three vectors you are left with lin…