(Solved):       1. Use the method of Example 3 to show that the following set of vectors ...

 

 

 

1. Use the method of Example 3 to show that the following set of vectors forms a basis for \( R^{2} \). \[ \{(2,1),(3,0)\} \] 2. Use the method of Example 3 to show that the following set of vectors forms a basis for \( R^{3} \). \[ \{(3,1,-4),(2,5,6),(1,4,8)\} \] 3. Show that the following polynomials form a basis for \( P_{2} \). \[ x^{2}+1, \quad x^{2}-1, \quad 2 x-1 \] 4. Show that the following polynomials form a basis for \( P_{3} \). \[ 1+x, \quad 1-x, \quad 1-x^{2}, \quad 1-x^{3} \] 5. Show that the following matrices form a basis for \( M_{22} \). \[ \left[\begin{array}{rr} 3 & 6 \\ 3 & -6 \end{array}\right], \quad\left[\begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array}\right], \quad\left[\begin{array}{rr} 0 & -8 \\ -12 & -4 \end{array}\right], \quad\left[\begin{array}{rr} 1 & 0 \\ -1 & 2 \end{array}\right] \] 6. Show that the following matrices form a basis for \( M_{22} \). \[ \left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right], \quad\left[\begin{array}{rr} 1 & -1 \\ 0 & 0 \end{array}\right], \quad\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right], \quad\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right] \] 7. In each part, show that the set of vectors is not a basis for \( R^{3} \). (a) \( \{(2,-3,1),(4,1,1),(0,-7,1)\} \) (b) \( \{(1,6,4),(2,4,-1),(-1,2,5)\} \) 8. Show that the following vectors do not form a basis for \( P_{2} \). \[ 1-3 x+2 x^{2}, \quad 1+x+4 x^{2}, \quad 1-7 x \] 1. Let \( V \) be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on \( \mathbf{u}=\left(u_{1}, u_{2}\right) \) and \( \mathbf{v}=\left(v_{1}, v_{2}\right) \) : \[ \mathbf{u}+\mathbf{v}=\left(u_{1}+v_{1}, u_{2}+v_{2}\right), \quad k \mathbf{u}=\left(0, k u_{2}\right) \] (a) Compute \( \mathbf{u}+\mathbf{v} \) and \( k \mathbf{u} \) for \( \mathbf{u}=(-1,2), \mathbf{v}=(3,4) \), and \( k=3 \). (b) In words, explain why \( V \) is closed under addition and scalar multiplication. (c) Since addition on \( V \) is the standard addition operation on \( R^{2} \), certain vector space axioms hold for \( V \) because they are known to hold for \( R^{2} \). Which axioms are they? (d) Show that Axioms 7, 8, and 9 hold. 1. Use the method of Example 3 to show that the following set of vectors forms a basis for \( R^{2} \). \[ \{(2,1),(3,0)\} \] 2. Use the method of Example 3 to show that the following set of vectors forms a basis for \( R^{3} \). \[ \{(3,1,-4),(2,5,6),(1,4,8)\} \] 3. Show that the following polynomials form a basis for \( P_{2} \). \[ x^{2}+1, \quad x^{2}-1, \quad 2 x-1 \] 4. Show that the following polynomials form a basis for \( P_{3} \). \[ 1+x, \quad 1-x, \quad 1-x^{2}, \quad 1-x^{3} \] 5. Show that the following matrices form a basis for \( M_{22} \). \[ \left[\begin{array}{rr} 3 & 6 \\ 3 & -6 \end{array}\right], \quad\left[\begin{array}{rr} 0 & -1 \\ -1 & 0 \end{array}\right], \quad\left[\begin{array}{rr} 0 & -8 \\ -12 & -4 \end{array}\right], \quad\left[\begin{array}{rr} 1 & 0 \\ -1 & 2 \end{array}\right] \] 6. Show that the following matrices form a basis for \( M_{22} \). \[ \left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right], \quad\left[\begin{array}{rr} 1 & -1 \\ 0 & 0 \end{array}\right], \quad\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right], \quad\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right] \] 7. In each part, show that the set of vectors is not a basis for \( R^{3} \). (a) \( \{(2,-3,1),(4,1,1),(0,-7,1)\} \) (b) \( \{(1,6,4),(2,4,-1),(-1,2,5)\} \) 8. Show that the following vectors do not form a basis for \( P_{2} \). \[ 1-3 x+2 x^{2}, \quad 1+x+4 x^{2}, \quad 1-7 x \]