Page 510 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Let be the orthogonal projection of onto a subspace W.
16.

(a) Prove that .

(b) What does the result in part (a) imply about the composition ?
(c) Show that is symmetric.
(d) Verify that the matrices in Tables 4 and 5 of Section 4.2 have the properties in parts (a) and (c).

Let A be an matrix with linearly independent row vectors. Find a standard matrix for the orthogonal projection of

17. onto the row space of A.

Hint Start with Formula 6.

The relationship between the current I through a resistor and the voltage drop V across it is given by Ohm's Law .

18. Successive experiments are performed in which a known current (measured in amps) is passed through a resistor of

unknown resistance R and the voltage drop (measured in volts) is measured. This results in the data (0.1, 1), (0.2,

2.1), (0.3, 2.9), (0.4, 4.2), (0.5, 5.1). The data is assumed to have measurement errors that prevent it from following

Ohm's Law precisely.

(a) Set up a linear system that represents the 5 equations , …, .

(b) Is this system consistent?

(c) Find the least squares solution of this system and interpret your result.

Repeat Exercise 18 under the assumption that the relationship between the current I and the voltage drop V is best

19. modeled by an equation of the form , where c is a constant offset value. This leads to a linear system.

Use the techniques of Section 4.4 to fit a polynomial of degree 4 to the data of Exercise 18. Is there a physical
20. interpretation of your result?

The following is the proof that in Theorem 6.4.3. Justify each line by filling in the
21. blank appropriately.

Hypothesis: Suppose that A is an matrix and is invertible.

Conclusion: A has linearly independent column vectors.