Vectors and linear combinations

1. Given the vectors , can $\bvec$ be expressed as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$? Rephrase this question by writing a linear system for the weights $c_1$, $c_2$, and $c_3$ and use the Sage cell below to answer this question.

2. Consider the following linear system. Identify vectors $\vvec_1$, $\vvec_2$, $\vvec_3$, and $\bvec$ such that the question "Is this linear system consistent?" is equivalent to the question "Can $\bvec$ be expressed as a linear…

Consider the vectors

1. Find the linear combination with weights $c_1 = 2$, $c_2=-3$, and $c_3=1$.

2. Can you write the vector ${\mathbf 0} = \left[\begin{array}{r} 0 \\ 0 \end{array}\right]$ as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$? If so, describe all the ways in which you can do so.

3. Can you write the vector ${\mathbf 0} = \left[\begin{array}{r} 0 \\ 0 \end{array}\right]$ as a linear combination using just the first two vectors $\vvec_1$ $\vvec_2$? If so, describe all the ways in which you can do so.

4. Can you write $\vvec_3$ as a linear combination of $\vvec_1$ and $\vvec_2$? If so, in how…

Nutritional information about a breakfast cereal is printed on the box. For instance, one serving of Frosted Flakes has 111 calories, 140 milligrams of sodium, and 1.2 grams of protein. We may represent this as a vector . One serving of Cocoa Puffs has 120 calories, 105 milligrams of sodium, and 1.0 grams of protein.

1. Write the vector describing the nutritional content of Cocoa Puffs.

2. Suppose you eat…

Consider the vectors

1. Can you express the vector $\bvec=\left[\begin{array}{r} 10 \\ 1 \\ -8 \end{array}\right]$ as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$? If so, describe all the ways in which you can do so.

2. Can you express the vector $\bvec=\left[\begin{array}{r} 3 \\ 7 \\ 1 \end{array}\right]$ as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$? If so, describe all the ways in which you can do so.

3. Show that $\vvec_3$ can be written as a linear combination of $\vvec_1$ and $\vvec_2$.

4. Explain why any linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$, can be rewritten as a l…

Consider the vectors For what value(s) of $k$, if any, can the vector $\left[\begin{array}{r} k \\ -2 \\ 5 \end{array}\right]$ be written as a linear combination of $\vvec_1$ and $\vvec_2$?

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

1. Given two vectors $\vvec$ and $\wvec$, the vector $2\vvec$ is a linear combination of $\vvec$ and $\wvec$.

2. Suppose $\vvec_1,\vvec_2,\ldots,\vvec_n$ is a collection of $m$-dimensional vectors and that the matrix $\left[\begin{array}{rrrr} \vvec_1 & \vvec_2 & \ldots & \vvec_n \end{array}\right]$ has a pivot position in every row. If $\bvec$ is any $m$-dimensional vector, then $\bvec$ can be written as a linear combination of $\vvec_1,\vvec_2,\ldots,\vvec_n$.

3. Suppose $\vvec_1,\vvec_2,\ldots,\vvec_n$ is a collection…