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Linear Algebra: Vector Space Definition Quiz

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Linear Algebra: Vector Space Definition Quiz

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Questions and Answers

What is the definition of a vector in the plane?

A matrix of real numbers

A single real number

A combination of real and imaginary numbers

An ordered pair of real numbers (correct)

What does a vector in the plane have?

Magnitude only

Direction only

Neither magnitude nor direction

Magnitude and direction (correct)

In the context of vectors, when are two matrices X and Y equal?

$X = Y$ if $x_1 = y_1$ and $x_2 = y_2$ (correct)

$X = Y$ if $x_1 imes y_1 = x_2 imes y_2$

$X = Y$ if $x_1 + y_1 = x_2 + y_2$

$X = Y$ if $x_1 - y_1 = x_2 - y_2$

What is the definition of an n-vector?

An n × 1 matrix Signup and view all the answers

What is the result of vector addition u + v for two vectors u and v?

$(u_1 + v_1, u_2 + v_2, …, u_n + v_n)$ Signup and view all the answers

What is the norm or length of the vector 𝑢 = (1, -2, -4, 5, 3)?

55 Signup and view all the answers

In vector addition and scalar multiplication, which property states that 𝒖 + (– 𝒖) = 𝟎?

Additive inverse property Signup and view all the answers

Which property in vector addition and scalar multiplication states that (𝑐 + 𝑑)𝒖 = 𝑐𝒖 + 𝑑𝒖?

Distributive property of scalar multiplication over real-number addition Signup and view all the answers

What is the unique unit vector in the same direction as v if v = (1, -3, 4, 2)?

(1, -1, 5, 1) Signup and view all the answers

If 𝒖 = (𝒖₁, 𝒖₂, … , 𝒖ₙ) and 𝒗 = (𝒗₁, 𝒗₂, … , 𝒗ₙ), what is the result of the expression 𝒖 + 𝒗?

(𝒖₁ + 𝒗₁ , 𝒖₂ + 𝒗₂ , … , 𝒖ₙ + 𝒗ₙ) Signup and view all the answers