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.
(c)
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.
Sketch the vectors \(\vvec, \wvec, \vvec + \wvec\) in the \(xy\) plane.
(b)
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 \(\wvec\text{.}\)
a line through the tip of \(\wvec\) parallel to \(\vvec\text{.}\)
a line through \((0,0)\) parallel to \(\vvec+\wvec\text{.}\)
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 \(b = 1\text{.}\)
(b)
Can the vector \(\left[\begin{array}{r} -31 \\ 37
\end{array}\right]\) be represented as a linear combination of \(\vvec\) and \(\wvec\text{?}\)
Activity2.1.0.4.
In this activity, we will look at linear combinations of a pair of vectors,
Go to the Applet in Activity 2.1.2 2 in Understanding Linear Algebra.
(a)
The weight \(b\) is initially set to 0. Explain what happens as you vary \(a\) with \(b=0\text{?}\) How is this related to scalar multiplication?
(b)
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.
(c)
The vectors that arise when the weight \(b\) is set to 1 and \(a\) is varied is
a line through \(\vvec\) parallel to \(\wvec\text{.}\)
a line through \(\wvec\) parallel to \(\vvec\text{.}\)
a line through \((0,0)\) parallel to \(\vvec+\wvec\text{.}\)
(d)
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 \(b\text{?}\)
(e)
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 \(b\text{?}\)
(f)
Verify the result from the previous part by algebraically finding the weights \(a\) and \(b\) that form the linear combination \(\left[\begin{array}{r} 3 \\ 0 \end{array} \right]\text{.}\)
(g)
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 \(\left[\begin{array}{r} 15.2 \\ 7.1 \end{array} \right]\text{?}\)
(h)
Are there any two-dimensional vectors that cannot be expressed as linear combinations of \(\vvec\) and \(\wvec\text{?}\)
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 \(\vvec_3\text{?}\)"
Which statement is the most accurate for \(\bvec\text{?}\)
The vector \(\bvec\) can be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\) in exactly one way.
The vector \(\bvec\) can be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\) in more than one way.
The vector \(\bvec\) cannot be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\text{.}\)
(b)
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.
There are vectors \(\bvec\) that cannot be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\text{.}\)
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.
Which statement is the most accurate for \(\bvec\text{?}\)
The vector \(\bvec\) can be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\) in exactly one way.
The vector \(\bvec\) can be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\) in more than one way.
The vector \(\bvec\) cannot be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\text{.}\)
(b)
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.
There are vectors \(\bvec\) that cannot be expressed as a linear combination of \(\vvec_1\text{,}\)\(\vvec_2\text{,}\) and \(\vvec_3\text{.}\)
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.
