By the end of this module, you should be able to
Multivariate data:
Why place data into a matrix?
That is, the data appears reduced to three rows.
This will turn out to be an effective way to reduce the dimension of data sets.
Consider this example:
In-Class Exercise 1: How does one address the question for the above example?
In-Class Exercise 2: Can {\bf u} be expressed in terms of {\bf v} and {\bf w}?
In-Class Exercise 3: Show that this is the case.
In-Class Exercise 4: Can {\bf u} above be expressed as a linear combination of {\bf v} and {\bf z}? Or {\bf v} as a linear combination of {\bf u} and {\bf z}?
We are led to a definition of linear independence:
Another example:
Recall (Proposition 5.7):
Now we can connect the two:
What does this tell us when there is dependence among the vectors?
As a preliminary step, let's prove the following about any RREF's rows and columns:
In-Class Exercise 5: Use the definition of linear independence to show both parts of Proposition 6.1 are true for this example, and then generalize to prove 6.1 for all RREFs.
In-Class Exercise 6: Use the definition of linear independence to show both parts of Proposition 6.2 are true for this example, and then generalize to prove 6.2 for all RREFs.
Next, let's look at an important idea: rank
In-Class Exercise 7: Show that the above is true given what was shown earlier for {\bf v}_2 and {\bf v}_4.
In-Class Exercise 8: Prove the following: if the vectors {\bf v}_1, {\bf v}_2, \ldots, {\bf v}_n are independent, then so is any subset of these vectors.
And now for two remarkable results:
Let us revisit an idea we've seen before: span
In-Class Exercise 9: In 3D, explore the span of {\bf u}=(1,1,2), {\bf v}=(2,1,3), {\bf w}=(3,1,4) in Span3DExample.java. You will need your MatrixTool.java from earlier, or you can use your implementation of LinTool to compute scalar multiplication and addition of vectors.
In-Class Exercise 10: We'll now explore the span of just two of the above vectors: {\bf u}=(1,1,2), {\bf v}=(2,1,3) in Span3DExample2.java. Is the span of the two the same as the span of the three? Try different bounds for the scalars. What is the span of just {\bf u}=(1,1,2) all by itself, and is that the same as any of the other spans?
In-Class Exercise 11: On paper, draw the vectors {\bf u}=(1,1,0), {\bf v}=(-1,1,0), {\bf w}=(0,-1,0). What is the set of vectors spanned by these three vectors? What two obvious vectors should one use to span the same set?
We will now define an accompanying term: subspace
This leads to the next important concept: dimension of a subspace
In-Class Exercise 12: Why?
Now for an interesting result:
In-Class Exercise 13: Show that {\bf u}, {\bf v} are independent and that {\bf w} is a linear combination of {\bf u} and {\bf v}.
In-Class Exercise 14: Prove Proposition 6.5. Hint: start by assuming that one vector among {\bf v}_1, {\bf v}_2, \ldots, {\bf v}_n is dependent on the others. Can the span be generated by the others?
With this background, we are in a position to prove a remarkable result.
First, some definitions:
Now for a surprising and powerful result:
We will prove Theorem 6.6 in steps:
In-Class Exercise 15: Show that the rowspace of {\bf A}^\prime is the same as the rowspace of {\bf A}. Do this for each of three types of row operations.
In-Class Exercise 16: Why are the pivot rows independent?
In-Class Exercise 17: Complete the proof of Proposition 6.4.
Finally, let's reinterpret the familiar {\bf Ax}={\bf b}:
Is there an {\bf x}=(x_1,x_2,x_3) such that {\bf Ax}={\bf b}?is the same as the question
Are there scalars x_1,x_2,x_3 such that {\bf b} = x_1{\bf v}_1 + x_2{\bf v}_2 + x_3{\bf v}_3?
In-Class Exercise 18: Prove or disprove the following. Suppose {\bf v}_1, {\bf v}_2, \ldots, {\bf v}_n is a collection of m-dimensional linearly independent vectors and {\bf u} \in \mbox{span}({\bf v}_1, {\bf v}_2, \ldots, {\bf v}_n). Then there is exactly one linear combination of {\bf v}_1, {\bf v}_2, \ldots, {\bf v}_n that will produce {\bf u}. Consider both cases, m=n and m\neq n.
We've encountered this idea before, but let's revisit:
In-Class Exercise 19: What is the more natural basis (3D vectors) for the (x,y) plane?
Computing a basis:
Note the difference between rows and columns:
Let's connect dimension and basis directly:
Recall what it means for two vectors to be orthogonal:
Orthogonal vectors are linearly independent:
In-Class Exercise 20: Prove this result:
In-Class Exercise 21: Does the same proof work for complex vectors? Recall that the dot-product for complex vectors is a bit different. If the proof fails, why does it fail? If the proof works, explain all the steps in terms of complex vectors.
Let's summarize a few key points in this module:
Study tips: