Linear Algebra Ch 1.3 — Vector Equations

Compute \(\mathbf{u} + \mathbf{v}\) and \(\mathbf{u} - 2 \mathbf{v}\). \(\mathbf{u} = \vec{-1, 2}\), \(\mathbf{v} = \vec{-3, 3}\)

Compute \(\mathbf{u} + \mathbf{v}\) and \(\mathbf{u} - 2 \mathbf{v}\). \(\mathbf{u} = \vec{3, 2}\), \(\mathbf{v} = \vec{2, 3}\)

Display the following vectors using arrows on an \(x y\)-graph: \(\mathbf{u}\), \(\mathbf{v}\), \(-\mathbf{v}\), \(-2 \mathbf{v}\), \(\mathbf{u} + \mathbf{v}\), \(\mathbf{u} - \mathbf{v}\), and \(\mathbf{u} - 2 \mathbf{v}\). Notice that \(\mathbf{u} - \mathbf{v}\) is the vertex of a parallelogram whose other vertices are \(\mathbf{u}\), \(\mathbf{0}\), and \(-\mathbf{v}\). \(\mathbf{u}\) and \(\mathbf{v}\) as in Exercise 1.

Display the following vectors using arrows on an \(x y\)-graph: \(\mathbf{u}\), \(\mathbf{v}\), \(-\mathbf{v}\), \(-2 \mathbf{v}\), \(\mathbf{u} + \mathbf{v}\), \(\mathbf{u} - \mathbf{v}\), and \(\mathbf{u} - 2 \mathbf{v}\). \(\mathbf{u}\) and \(\mathbf{v}\) as in Exercise 2.

Write a system of equations that is equivalent to the given vector equation. \(x_1 \vec{4, -3, 2} + x_2 \vec{-8, 7, 0} = \vec{9, -6, -5}\)

Write a system of equations that is equivalent to the given vector equation. \(x_1 \vec{-2, 3} + x_2 \vec{8, 5} + x_3 \vec{1, -6} = \vec{0, 0}\)

Use the accompanying figure to write each vector listed as a linear combination of \(\mathbf{u}\) and \(\mathbf{v}\). Is every vector in \(RR^2\) a linear combination of \(\mathbf{u}\) and \(\mathbf{v}\)? Vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\), and \(\mathbf{d}\).

Use the accompanying figure to write each vector listed as a linear combination of \(\mathbf{u}\) and \(\mathbf{v}\). Vectors \(\mathbf{w}\), \(\mathbf{x}\), \(\mathbf{y}\), and \(\mathbf{z}\).

Write a vector equation that is equivalent to the given system of equations. \(x_2 + 5 x_3 = 0\) \(4 x_1 + 6 x_2 - x_3 = 0\) \(-x_1 + 3 x_2 - 8 x_3 = 0\)

Write a vector equation that is equivalent to the given system of equations. \(4 x_1 + x_2 + 3 x_3 = 9\) \(x_1 - 7 x_2 - 2 x_3 = 2\) \(8 x_1 + 6 x_2 - 5 x_3 = 15\)

Determine if \(\mathbf{b}\) is a linear combination of \(\mathbf{a}_1\), \(\mathbf{a}_2\), and \(\mathbf{a}_3\). \(\mathbf{a}_1 = \vec{1, -2, 0}\), \(\mathbf{a}_2 = \vec{0, 1, 2}\), \(\mathbf{a}_3 = \vec{5, -6, 8}\), \(\mathbf{b} = \vec{2, -1, 6}\)

Determine if \(\mathbf{b}\) is a linear combination of \(\mathbf{a}_1\), \(\mathbf{a}_2\), and \(\mathbf{a}_3\). \(\mathbf{a}_1 = \vec{1, -2, 2}\), \(\mathbf{a}_2 = \vec{0, 5, 5}\), \(\mathbf{a}_3 = \vec{2, 0, 8}\), \(\mathbf{b} = \vec{-5, 11, -7}\)

Determine if \(\mathbf{b}\) is a linear combination of the columns of the matrix \(A\). \(A = \begin{pmatrix} 1 & -4 & 2 \\ 0 & 3 & 5 \\ -2 & 8 & -4 \end{pmatrix}\), \(\mathbf{b} = \vec{3, -7, -3}\)

Determine if \(\mathbf{b}\) is a linear combination of the columns of the matrix \(A\). \(A = \begin{pmatrix} 1 & -2 & -6 \\ 0 & 3 & 7 \\ 1 & -2 & 5 \end{pmatrix}\), \(\mathbf{b} = \vec{11, -5, 9}\)

List five vectors in \(\text{Span} {\mathbf{v}_1, \mathbf{v}_2}\). For each vector, show the weights on \(\mathbf{v}_1\) and \(\mathbf{v}_2\) used to generate the vector and list the three entries of the vector. Do not make a sketch. \(\mathbf{v}_1 = \vec{7, 1, -6}\), \(\mathbf{v}_2 = \vec{-5, 3, 0}\)

List five vectors in \(\text{Span} {\mathbf{v}_1, \mathbf{v}_2}\). For each vector, show the weights on \(\mathbf{v}_1\) and \(\mathbf{v}_2\) used to generate the vector and list the three entries of the vector. Do not make a sketch. \(\mathbf{v}_1 = \vec{3, 0, 2}\), \(\mathbf{v}_2 = \vec{-2, 0, 3}\)

Let \(\mathbf{a}_1 = \vec{1, 4, -2}\), \(\mathbf{a}_2 = \vec{-2, -3, 7}\), and \(\mathbf{b} = \vec{4, 1, h}\). For what value(s) of \(h\) is \(\mathbf{b}\) in the plane spanned by \(\mathbf{a}_1\) and \(\mathbf{a}_2\)?

Let \(\mathbf{v}_1 = \vec{1, 0, -4}\), \(\mathbf{v}_2 = \vec{-5, 1, 7}\), and \(\mathbf{y} = \vec{h, -1, -5}\). For what value(s) of \(h\) is \(\mathbf{y}\) in the plane generated by \(\mathbf{v}_1\) and \(\mathbf{v}_2\)?

Give a geometric description of \(\text{Span} {\mathbf{v}_1, \mathbf{v}_2}\) for the vectors \(\mathbf{v}_1 = \vec{8, 2, -6}\) and \(\mathbf{v}_2 = \vec{12, 3, -9}\).

Give a geometric description of \(\text{Span} {\mathbf{v}_1, \mathbf{v}_2}\) for the vectors in Exercise 16.

Let \(\mathbf{u} = \vec{2, -1}\) and \(\mathbf{v} = \vec{2, 1}\). Show that \(\vec{h, k}\) is in \(\text{Span} {\mathbf{u}, \mathbf{v}}\) for all \(h\) and \(k\).

Construct a \(3 \times 3\) matrix \(A\), with nonzero entries, and a vector \(\mathbf{b}\) in \(RR^3\) such that \(\mathbf{b}\) is not in the set spanned by the columns of \(A\).

(T/F) Another notation for the vector \(\vec{-4, 3}\) is \([-4 \ 3]\).

(T/F) Any list of five real numbers is a vector in \(RR^5\).

(T/F) The points in the plane corresponding to \(\vec{-2, 5}\) and \(\vec{-5, 2}\) lie on a line through the origin.

(T/F) The vector \(\mathbf{u}\) results when a vector \(\mathbf{u} - \mathbf{v}\) is added to the vector \(\mathbf{v}\).

(T/F) An example of a linear combination of vectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\) is the vector \(\dfrac{1}{2} \mathbf{v}_1\).

(T/F) The weights \(c_1, ..., c_p\) in a linear combination \(c_1 \mathbf{v}_1 + \cdots + c_p \mathbf{v}_p\) cannot all be zero.

(T/F) The solution set of the linear system whose augmented matrix is \([\mathbf{a}_1 \ \mathbf{a}_2 \ \mathbf{a}_3 \ \mathbf{b}]\) is the same as the solution set of the equation \(x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + x_3 \mathbf{a}_3 = \mathbf{b}\).

(T/F) When \(\mathbf{u}\) and \(\mathbf{v}\) are nonzero vectors, \(\text{Span} {\mathbf{u}, \mathbf{v}}\) contains the line through \(\mathbf{u}\) and the origin.

(T/F) The set \(\text{Span} {\mathbf{u}, \mathbf{v}}\) is always visualized as a plane through the origin.

(T/F) Asking whether the linear system corresponding to an augmented matrix \([\mathbf{a}_1 \ \mathbf{a}_2 \ \mathbf{a}_3 \ \mathbf{b}]\) has a solution amounts to asking whether \(\mathbf{b}\) is in \(\text{Span} {\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3}\).

Let \(A = \begin{pmatrix} 1 & 0 & -4 \\ 0 & 3 & -2 \\ -2 & 6 & 3 \end{pmatrix}\) and \(\mathbf{b} = \vec{4, 1, -4}\). Denote the columns of \(A\) by \(\mathbf{a}_1\), \(\mathbf{a}_2\), \(\mathbf{a}_3\), and let \(W = \text{Span} {\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3}\). a. Is \(\mathbf{b}\) in \({\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3}\)? How many vectors are in \({\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3}\)? b. Is \(\mathbf{b}\) in \(W\)? How many vectors are in \(W\)? c. Show that \(\mathbf{a}_1\) is in \(W\). [Hint: Row operations are unnecessary.]

Let \(A = \begin{pmatrix} 2 & 0 & 6 \\ -1 & 8 & 5 \\ 1 & -2 & 1 \end{pmatrix}\), let \(\mathbf{b} = \vec{10, 3, 3}\), and let \(W\) be the set of all linear combinations of the columns of \(A\). a. Is \(\mathbf{b}\) in \(W\)? b. Show that the third column of \(A\) is in \(W\).

A mining company has two mines. One day's operation at mine 1 produces ore that contains 20 metric tons of copper and 550 kilograms of silver, while one day's operation at mine 2 produces ore that contains 30 metric tons of copper and 500 kilograms of silver. Let \(\mathbf{v}_1 = \vec{20, 550}\) and \(\mathbf{v}_2 = \vec{30, 500}\). a. What physical interpretation can be given to the vector \(5 \mathbf{v}_1\)? b. Suppose the company operates mine 1 for \(x_1\) days and mine 2 for \(x_2\) days. Write a vector equation whose solution gives the number of days each mine should operate in order to produce 150 tons of copper and 2825 kilograms of silver. Do not solve the equation. c. Solve the equation in (b).

A steam plant burns two types of coal: anthracite (A) and bituminous (B). For each ton of A burned, the plant produces 27.6 million Btu of heat, 3100 grams (g) of sulfur dioxide, and 250 g of particulate matter. For each ton of B burned, the plant produces 30.2 million Btu, 6400 g of sulfur dioxide, and 360 g of particulate matter. a. How much heat does the steam plant produce when it burns \(x_1\) tons of A and \(x_2\) tons of B? b. Suppose the output of the steam plant is described by a vector that lists the amounts of heat, sulfur dioxide, and particulate matter. Express this output as a linear combination of two vectors, assuming that the plant burns \(x_1\) tons of A and \(x_2\) tons of B. c. Over a certain time period, the steam plant produced 162 million Btu of heat, 23,610 g of sulfur dioxide, and 1623 g of particulate matter. Determine how many tons of each type of coal the steam plant must have burned. Include a vector equation as part of your solution.

Let \(\mathbf{v}_1, ..., \mathbf{v}_k\) be points in \(RR^3\) and suppose that for \(j = 1, ..., k\) an object with mass \(m_j\) is located at point \(\mathbf{v}_j\). Physicists call such objects point masses. The total mass of the system of point masses is \(m = m_1 + \cdots + m_k\) The center of mass (or center of gravity) of the system is \(macron(\mathbf{v}) = \dfrac{1}{m} [m_1 \mathbf{v}_1 + \cdots + m_k \mathbf{v}_k]\) Compute the center of gravity of the system consisting of the following point masses: \(\mathbf{v}_1 = (5, -4, 3)\), mass 2 g \(\mathbf{v}_2 = (4, 3, -2)\), mass 5 g \(\mathbf{v}_3 = (-4, -3, -1)\), mass 2 g \(\mathbf{v}_4 = (-9, 8, 6)\), mass 1 g

Let \(\mathbf{v}\) be the center of mass of a system of point masses located at \(\mathbf{v}_1, ..., \mathbf{v}_k\) as in Exercise 37. Is \(\mathbf{v}\) in \(\text{Span} {\mathbf{v}_1, ..., \mathbf{v}_k}\)? Explain.

A thin triangular plate of uniform density and thickness has vertices at \(\mathbf{v}_1 = (0, 1)\), \(\mathbf{v}_2 = (8, 1)\), and \(\mathbf{v}_3 = (2, 4)\), as in the figure below, and the mass of the plate is 3 g. a. Find the \((x, y)\)-coordinates of the center of mass of the plate. This "balance point" of the plate coincides with the center of mass of a system consisting of three 1-gram point masses located at the vertices of the plate. b. Determine how to distribute an additional mass of 6 g at the three vertices of the plate to move the balance point of the plate to \((2, 2)\). [Hint: Let \(w_1\), \(w_2\), and \(w_3\) denote the masses added at the three vertices, so that \(w_1 + w_2 + w_3 = 6\).]

Consider the vectors \(\mathbf{v}_1\), \(\mathbf{v}_2\), \(\mathbf{v}_3\), and \(\mathbf{b}\) in \(RR^2\), shown in the figure. Does the equation \(x_1 \mathbf{v}_1 + x_2 \mathbf{v}_2 + x_3 \mathbf{v}_3 = \mathbf{b}\) have a solution? Is the solution unique? Use the figure to explain your answers.

Use the vectors \(\mathbf{u} = (u_1, ..., u_n)\), \(\mathbf{v} = (v_1, ..., v_n)\), and \(\mathbf{w} = (w_1, ..., w_n)\) to verify the following algebraic properties of \(RR^n\). a. \((\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})\) b. \(c(\mathbf{u} + \mathbf{v}) = c \mathbf{u} + c \mathbf{v}\) for each scalar \(c\)

Use the vector \(\mathbf{u} = (u_1, ..., u_n)\) to verify the following algebraic properties of \(RR^n\). a. \(\mathbf{u} + (-\mathbf{u}) = (-\mathbf{u}) + \mathbf{u} = \mathbf{0}\) b. \(c(d \mathbf{u}) = (c d) \mathbf{u}\) for all scalars \(c\) and \(d\)