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GRE Subject Test: Math : Linear Algebra

Study concepts, example questions & explanations for GRE Subject Test: Math

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Example Questions

Example Question #1 : Vectors & Spaces

Calculate the dot product of the following vectors:

$a=\left \langle c,6, s\right \rangle$

$b=\left \langle c,10,s\right \rangle$

Possible Answers:

c^2+16+s^2

2cs+60

c^2+60+s^2

60c^2s^2

2c+16+2s

Correct answer:

c^2+60+s^2

Explanation:

Write the formula for dot product given $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ and $b=\left \langle b_{1},b_{2},b_{3}\right \rangle$ .

$a\cdot b= a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$

Substitute the values of the vectors to determine the dot product.

$a\cdot b= c(c)+(6)(10)+s(s) = c^2+60+s^2$

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Example Question #1 : Vector Form

Express -i-k in vector form.

Possible Answers:

$\left \langle -1,-1,0\right \rangle$

$\left \langle -1,-1\right \rangle$

$\left \langle -1,0,-1\right \rangle$

$\left \langle -1,1,-1\right \rangle$

$\left \langle 0,-1,-1\right \rangle$

Correct answer:

$\left \langle -1,0,-1\right \rangle$

Explanation:

The correct form of x,y, and z of a vector is represented in the order of i, j, and k, respectively. The coefficients of i,j, and k are used to write the vector form.

$-i-k = -1i+0j-1k = \left \langle -1,0,-1\right \rangle$

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Example Question #1 : Vector Form

Express -10j in vector form.

Possible Answers:

$\left \langle -10,0,0\right \rangle$

$\left \langle 0,-10,0\right \rangle$

$\left \langle 0,0,-10\right \rangle$

$\left \langle -10,-10,-10\right \rangle$

$\left \langle 1,-10,1\right \rangle$

Correct answer:

$\left \langle 0,-10,0\right \rangle$

Explanation:

The x,y, and z of a vector is represented in the order of i, j, and k, respectively. Use the coefficients of i,j, and k to write the vector form.

$-10j=0i-10j+0k = \left \langle 0,-10,0\right \rangle$

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Example Question #1 : Vector Form

Find the vector form of (6,3,1) to (0,1,3) .

Possible Answers:

$\overrightarrow{v}=[6,2,-3]$

$\overrightarrow{v}=[-6,-2,-2]$

$\overrightarrow{v}=[6,2,2]$

$\overrightarrow{v}=[6,-2,-2]$

$\overrightarrow{v}=[6,2,-2]$

Correct answer:

$\overrightarrow{v}=[6,2,-2]$

Explanation:

When we are trying to find the vector form we need to remember the formula which states to take the difference between the ending and starting point.

Thus we would get:

Given (a,b,c) and (d,e,f)

$\overrightarrow{v}=[d-a, e-b, f-c]$

In our case we have ending point at (6,3,1) and our starting point at (0,1,3) .

Therefore we would set up the following and simplify.

$\overrightarrow{v}=[6-0,3-1,1-3]=[6,2,-2]$

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Example Question #25 : Linear Algebra

Find the dot product of the 2 vectors. $\left \langle a,2a\right \rangle \cdot \left \langle a,a\right \rangle$

Possible Answers:

$\left \langle a^2,2a^2\right \rangle$

$\left \langle 3a^2,3a^2\right \rangle$

$\left \langle a,2a\right \rangle$

3a^2

Correct answer:

3a^2

Explanation:

The dot product will give a single value answer, and not a vector as a result.

To find the dot product, use the following formula:

$\left \langle x_1,x_2\right \rangle \cdot \left \langle y_1,y_2\right \rangle=(x_1)(y_1)+(x_2)(y_2)$

$\left \langle a,2a\right \rangle \cdot \left \langle a,a\right \rangle = (a)(a)+(2a)(a)=a^2+2a^2=3a^2$

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Example Question #2 : Vectors & Spaces

Assume that Billy fired himself out of a circus cannon at a velocity of $25 \:mph$ at an elevation angle of degrees. Write this in vector component form.

Possible Answers:

$\left \langle22.658,10.565 \right \rangle$

$\left \langle 25.25,25.25\right \rangle$

$\left \langle23.097, 9.567 \right \rangle$

$\left \langle 50,50\right \rangle$

$\left \langle 24.78,-3.309\right \rangle$

Correct answer:

$\left \langle22.658,10.565 \right \rangle$

Explanation:

The firing of the cannon has both x and y components.

Write the formula that distinguishes the x and y direction and substitute.

$\left \langle x,y\right \rangle= \left \langle vcos(\theta),vsin(\theta)\right \rangle$

Ensure that the calculator is in degree mode before you solve.

$\left \langle 25cos25,25sin25\right \rangle = \left \langle22.658,10.565 \right \rangle$

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Example Question #3 : Vectors & Spaces

Compute: $2\vec{a}-6\vec{b}$ given the following vectors. $\vec{a}= \left \langle 1,3\right \rangle$ and $\vec{b}= \left \langle 1\right \rangle$ .

Possible Answers:

$\left \langle -4,0\right \rangle$

$\left \langle -4,3\right \rangle$

${}\left< \sqrt{13} \right \rangle$

The answer does not exist.

$\left \langle 1,2\right \rangle$

Correct answer:

The answer does not exist.

Explanation:

The dimensions of the vectors are mismatched.

Since vector $\vec{a}$ does not have the same dimensions as $\vec{b}$ , the answer for $2\vec{a}-6\vec{b}$ cannot be solved.

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Example Question #1 : Vector Form

What is the vector form of 2i-10k ?

Possible Answers:

$\left \langle 2,0,-10\right \rangle$

$\left \langle -10,2,0\right \rangle$

$\left \langle 2,-10,0\right \rangle$

Correct answer:

$\left \langle 2,0,-10\right \rangle$

Explanation:

To find the vector form of 2i-10k , we must map the coefficients of , , and to their corresponding , , and coordinates. Thus, 2i-10k becomes $\left \langle 2,0,-10\right \rangle$ .

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Example Question #1 : Vector Form

Express -2i+7j-10k in vector form.

Possible Answers:

$\left \langle 2,-7,10\right \rangle$

$\left \langle -2,-7,-10\right \rangle$

$\left \langle 2,7,-10\right \rangle$

$\left \langle -2,7,-10\right \rangle$

$\left \langle 2,7,10\right \rangle$

Correct answer:

$\left \langle -2,7,-10\right \rangle$

Explanation:

In order to express -2i+7j-10k in vector form, we must use the coefficients of i, j, and to represent the -, -, and -coordinates of the vector.

Therefore, its vector form is

$\left \langle -2,7,-10\right \rangle$ .

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Example Question #1 : Vector Form

Express i+5k in vector form.

Possible Answers:

$\left \langle -1,5\right \rangle$

$\left \langle -1,-5\right \rangle$

$\left \langle -1,0,5\right \rangle$

$\left \langle 1,5\right \rangle$

$\left \langle 1,0,5\right \rangle$

Correct answer:

$\left \langle 1,0,5\right \rangle$

Explanation:

In order to express i+5k in vector form, we must use the coefficients of i, j, and to represent the -, -, and -coordinates of the vector.

Therefore, its vector form is

$\left \langle 1,0,5\right \rangle$ .

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GRE Subject Test: Math : Linear Algebra

Study concepts, example questions & explanations for GRE Subject Test: Math

All GRE Subject Test: Math Resources

Example Questions

Example Question #1 : Vectors & Spaces

Example Question #1 : Vector Form

Example Question #1 : Vector Form

Example Question #1 : Vector Form

Example Question #25 : Linear Algebra

Example Question #2 : Vectors & Spaces

Example Question #3 : Vectors & Spaces

Example Question #1 : Vector Form

Example Question #1 : Vector Form

Example Question #1 : Vector Form

All GRE Subject Test: Math Resources

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