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Homework 1
- Solution Manual: Chapter 1; Chapter 2
- Section 1.1
- Watch Lecture 1: The geometry of linear equations of the MIT OpenCourseWare series
- A Geogebra linear combination activity
- In class: 9* (draw all three paralellograms):
- 9*. Follow the directions for Problem 9 in Problem Set 1.1, with the three points \((2,1), (1,3)\ \mbox{and}\ (3,2)\). Draw all three parallelograms.
- Homework: 1, 3, 5, 6, 8, 9, 11, 13, 15, 16, 20, 27, 31
- Hand In: 5*, 8*, 9**:
- 5*. Complete Problem 5 in Problem Set 1.1, but with the vectors \[{\bf u} = \left[\matrix{2\cr -3\cr 1\cr}\right], {\bf v} = \left[\matrix{-2\cr 3\cr -1\cr}\right]\ \mbox{and}\ {\bf w} = \left[\matrix{2\cr -5\cr -1\cr}\right]\]
- 8*. The parallelogram on the right of Figure 1.1 (made with vectors \({\bf v}\) and \(-{\bf w}\)) has diagonal \({\bf v}-{\bf w}\). What is its other diagonal? What is the sum of the two diagonals? (There are two possible answers for each part, but they must be consistent.)
- 9**. Follow the directions for Problem 9 in Problem Set 1.1, with the three points \((-1,1), (2,1)\ \mbox{and}\ (1,3)\). Draw all three parallelograms.
- Section 1.2
- In class: 7*:
- 7*. Find the angle \(\theta\) (from its cosine) between these pairs of vectors:
- (a) \({\bf v} = \left[\matrix{\sqrt{3}\cr -1\cr}\right]\) and \({\bf w} = \left[\matrix{1\cr 0\cr}\right]\)
- (b) \({\bf v} = \left[\matrix{1\cr 2\cr 2\cr}\right]\) and \({\bf w} = \left[\matrix{-2\cr 1\cr -2\cr}\right]\)
- (c) \({\bf v} = \left[\matrix{1\cr 3\cr}\right]\) and \({\bf w} = \left[\matrix{1\cr 2\cr}\right]\)
- 7*. Find the angle \(\theta\) (from its cosine) between these pairs of vectors:
- Homework: 1, 2, 4, 7, 9, 13, 15, 28
- Hand In: 7**, 13*, 28*:
- 7**. Find the angle \(\theta\) (from its cosine) between these pairs of vectors:
- (a) \({\bf v} = \left[\matrix{-\sqrt{3}\cr -1\cr}\right]\) and \({\bf w} = \left[\matrix{1\cr 0\cr}\right]\)
- (b) \({\bf v} = \left[\matrix{1\cr 2\cr -2\cr}\right]\) and \({\bf w} = \left[\matrix{-2\cr 1\cr 2\cr}\right]\)
- (c) \({\bf v} = \left[\matrix{1\cr -3\cr}\right]\) and \({\bf w} = \left[\matrix{1\cr 2\cr}\right]\)
- 13*. Find nonzero vectors \({\bf v}\) and \({\bf w}\) that are perpendicular to \((0,1, -1)\) and to each other.
- 28*. If \({\bf v} = (3,2)\) draw all vectors \({\bf w} = (x,y)\) in the \(xy\)-plane with \({\bf v}\cdot{\bf w} = 6\). Why do all those \({\bf w}\)'s lie along a line? Which is the shortest such \({\bf w}\)?
- 7**. Find the angle \(\theta\) (from its cosine) between these pairs of vectors:
- In class: 7*:
- Section 1.3
- In class: 6*:
- 6*. For each matrix determine which number(s) \(c\) give dependent columns and give a combination of columns that equals the zero vector:
- (a) \(\left[\matrix{2&1&1\cr 4&2&2\cr 8&3&c}\right]\)
- (b) \(\left[\matrix{0&1&c\cr 1&0&-1\cr 1&1&0}\right]\)
- (c) \(\left[\matrix{c&c&c\cr 1&1&1\cr 2&3&1\cr}\right]\)
- 6*. For each matrix determine which number(s) \(c\) give dependent columns and give a combination of columns that equals the zero vector:
- Homework: 2, 3, 4, 5, 6 (this is three different problems in one), 7 , 10
- Hand In: 6**, 10*:
- 6**. For each matrix determine which number(s) \(c\) give dependent columns and give a combination of columns that equals the zero vector:
- (a) \(\left[\matrix{1&1&0\cr 4&1&3\cr 6&2&c}\right]\)
- (b) \(\left[\matrix{1&1&0\cr 0&1&c\cr 1&0&-1}\right]\)
- (c) \(\left[\matrix{c&c&c\cr 2&2&2\cr 1&3&-3\cr}\right]\)
- 10*. Follow the directions for Problem 10 in Problem Set 1.3, but with the matrix \[\Delta = \left[\matrix{1&1&0\cr 0&-1&-1\cr 0&0&1\cr}\right].\]
- 6**. For each matrix determine which number(s) \(c\) give dependent columns and give a combination of columns that equals the zero vector:
- In class: 6*:
- Section 2.1
- A Geogebra system of three equations in three unknowns activity
- In class: 19*:
- 19*. What \(3 \times 3\) matrix \(E\) multiplies \((x,y,z)\) to give \((x,y+x,z)\)? What matrix \(E^{-1}\) multiplies \((x,y,z)\) to give \((x,y-x,z)\)? If you multiply \((1,-2,3)\) by \(E\) and then by \(E^{-1}\), the two results are ...?
- Homework: 1-4, 7-10, 13, 15-19, 22, 24, 25, 28-31, 33, 35
- Hand In: 7*, 17*, 19**:
- 7*. The first of these two equations plus the second equals the third: \[\matrix{x&-&y&+&z&=&4\cr x&+&2y&-&2z&=&-2\cr 2x&+&y&-&z&=&2}\] The columns are \((1,1,2), (-1,2,1)\) and \((1,-2,-1)\). This is a "singular case" because the third column is what linear combination of the first two columns? Find two different combinations of the three columns that give \({\bf b} = (4,-2,2)\). The system can be solved for \({\bf b} = (-4,2,c)\) only if \(c\) has what value?
- 17*. Find the matrix \(P\) that multiplies \((x,y,z)\) to give \((z,x,y)\). Find the matrix \(Q\) that multiplies \((z,x,y)\) to give \((x,y,z)\).
- 19**. What \(3 \times 3\) matrix \(E\) multiplies \((x,y,z)\) to give \((x,y,z+y)\)? What matrix \(E^{-1}\) multiplies \((x,y,z)\) to give \((x,y,z-y)\)? If you multiply \((4,2,-1)\) by \(E\) and then by \(E^{-1}\), the two results are ...?
- Section 2.2
- In class: 13*:
- 13*. Follow the directions for Problem 13 in Problem Set 2.2, with the system \[\matrix{x&+&2y&-&z&=&-3\cr 3x&+&5y&\ &\ &=&-9\cr 4x&+&11y&-&11z&=&-14}\]
- Homework: 1, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 17, 19, 21, 25, 27, 28
- Hand In: 7*, 13**, 21*:
- 7*. Follow the directions for Problem 7 in Problem Set 2.2, with the system \[\matrix{ax&+&2y&=&2\cr -3x&-&6y&=&6}\]
- 13**. Follow the directions for Problem 13 in Problem Set 2.2, with the system \[\matrix{2x&-&3y&+&z&=&-11\cr 4x&-&2y&+&z&=&-11\cr 8x&+&4y&+&3z&=&-9}\]
- 21*. Follow the directions for Problem 21 in Problem Set 2.2, with the systems \[\matrix{3x&+&y&\ &\ &\ &\ &=&0\cr x&+&3y&+&z&\ &\ &=&0\cr \ &\ &y&+&3z&+&t&=&0\cr \ &\ &\ &\ &z&+&3t&=&55}\] and \[\matrix{3x&-&y&\ &\ &\ &\ &=&0\cr -x&+&3y&-&z&\ &\ &=&0\cr \ &\ &-y&+&3z&-&t&=&0\cr \ &\ &\ &\ &-z&+&3t&=&55}\]
- In class: 13*: