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GATE MA — free MCQ practice

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GATE MA subjects

Mathematics (26)

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Linear Algebra

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A VECTOR SPACE $V$ over a field $\mathbb{F}$ requires which axiom about scalar multiplication?Read the answer and worked solution →A set of vectors $\{v_1, \ldots, v_n\}$ is LINEARLY INDEPENDENT iffRead the answer and worked solution →A BASIS of a vector space $V$ is a set that isRead the answer and worked solution →The DIMENSION of $\mathbb{R}^n$ isRead the answer and worked solution →

Worked practice questions

A VECTOR SPACE $V$ over a field $\mathbb{F}$ requires which axiom about scalar multiplication?

  1. $\alpha(\beta v) = (\alpha\beta)v$
  2. $\alpha v = v$ for every $\alpha$
  3. Scalars must be integers
  4. Vectors must be linearly independent
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Correct answer: A. $\alpha(\beta v) = (\alpha\beta)v$

1. A vector space has axioms for addition and scalar multiplication. 2. Scalar mult axiom: $\alpha(\beta v) = (\alpha\beta) v$ — associativity of scalar action. 3. Other scalar mult axioms: $1 \cdot v = v$, $\alpha(u + v) = \alpha u + \alpha v$, $(\alpha + \beta) v = \alpha v + \beta v$. 4. Option B has the WRONG identity (would force every vector to equal itself for ANY scalar). Options C, D are unrelated. _Source: Sergei Treil, "Linear Algebra Done Wrong", §1.1 (Vector spaces — axioms)._

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A set of vectors $\{v_1, \ldots, v_n\}$ is LINEARLY INDEPENDENT iff

  1. $\sum a_i v_i = 0$ forces all $a_i = 0$
  2. their dot products are all zero
  3. no vector equals zero
  4. $\sum v_i \neq 0$
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Correct answer: A. $\sum a_i v_i = 0$ forces all $a_i = 0$

1. DEFINITION: vectors are LINEARLY INDEPENDENT iff the only way to write $\sum a_i v_i = 0$ is with all coefficients $a_i = 0$. 2. Equivalently: no vector in the set is a linear combination of the others. 3. If some non-trivial $\sum a_i v_i = 0$ exists, the vectors are LINEARLY DEPENDENT. 4. Option B describes ORTHOGONALITY (different property). Options C, D are too weak. _Source: Sergei Treil, "Linear Algebra Done Wrong", §2.2 (Linear independence)._

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A BASIS of a vector space $V$ is a set that is

  1. orthogonal only
  2. the set of all eigenvectors
  3. infinite
  4. linearly independent AND spans $V$
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Correct answer: D. linearly independent AND spans $V$

1. A BASIS is a SET that is (a) LINEARLY INDEPENDENT and (b) SPANS the whole space $V$. 2. Every vector $v \in V$ has a UNIQUE expansion as a linear combination of basis vectors. 3. The number of vectors in a basis is the DIMENSION of $V$ — independent of which basis you choose. 4. Option A is for orthogonal bases (special case). Options C, D are not necessary properties. _Source: Sergei Treil, "Linear Algebra Done Wrong", §2.3 (Basis)._

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