What geometric effect does scalar multiplication by a positive number have on a vector? (A) preserves the length, changes the direction, (B) preserves the direction, changes the length, (C) changes the length and the direction.
What geometric effect does scalar multiplication by a negative number have on a vector? (A) preserves the length, changes the direction, (B) preserves the direction, changes the length, (C) changes the length and the direction.
Consider vectors that have the form \(\vvec +
a\wvec\) where \(a\) is any scalar. Sketch a few of these vectors when, say, \(a = -2, -1, 0, 1, \) and \(2\text{.}\) Consider the points at the tips of each of these vectors. An appropriate geometric description for this set of points would be
a line through the tip of \(\vvec\) parallel to \(\wvec\text{.}\)
If \(a\) and \(b\) are two scalars, then the vector
\begin{equation*}
a \vvec + b \wvec
\end{equation*}
is called a linear combination of the vectors \(\vvec\) and \(\wvec\text{.}\) Find the vector that is the linear combination when \(a = -2\) and \(b = 1\text{.}\)
What is the linear combination of \(\vvec\) and \(\wvec\) when \(a = 1\) and \(b=-2\text{?}\) You may find this result using the diagram, but you should also verify it by computing the linear combination.
Can the vector \(\left[\begin{array}{r} 0 \\ 0 \end{array} \right]\) be expressed as a linear combination of \(\vvec\) and \(\wvec\text{?}\) If so, what are weights \(a\) and \(b\text{?}\)
Can the vector \(\left[\begin{array}{r} 3 \\ 0 \end{array} \right]\) be expressed as a linear combination of \(\vvec\) and \(\wvec\text{?}\) If so, what are weights \(a\) and \(b\text{?}\)
Verify the result from the previous part by algebraically finding the weights \(a\) and \(b\) that form the linear combination \(\left[\begin{array}{r} 3 \\ 0 \end{array} \right]\text{.}\)
Can the vector \(\left[\begin{array}{r} 1.3 \\ -1.7 \end{array} \right]\) be expressed as a linear combination of \(\vvec\) and \(\wvec\text{?}\) What about the vector \(\left[\begin{array}{r} 15.2 \\ 7.1 \end{array} \right]\text{?}\)
we ask if \(\bvec\) can be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\text{.}\) Rephrase this question by writing a linear system for the weights \(c_1\text{,}\)\(c_2\text{,}\) and \(c_3\) and use the Sage cell below to answer this question.
Identify vectors \(\vvec_1\text{,}\)\(\vvec_2\text{,}\)\(\vvec_3\text{,}\) and \(\bvec\) and rephrase the question "Is this linear system consistent?" by asking "Can \(\bvec\) be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\text{?}\)"
Considering just the vectors \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\text{,}\) determine the most accurate statement.
Every three dimensional vector \(\bvec\) can be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\) in exactly one way.
Every three dimensional vector \(\bvec\) can be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\) in more than one way.
Be able to explain how the pivot positions of the matrix \(\left[\begin{array}{rrr} \vvec_1 \amp
\vvec_2 \amp \vvec_3 \end{array} \right]\) help answer this question.
Considering just the vectors \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\) from the previous part, determine the most accurate statement.
Every three dimensional vector \(\bvec\) can be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\) in exactly one way.
Every three dimensional vector \(\bvec\) can be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\) in more than one way.
Be able to explain how the pivot positions of the matrix \(\left[\begin{array}{rrr} \vvec_1 \amp
\vvec_2 \amp \vvec_3 \end{array} \right]\) help answer this question.
