Gani — AI Math Tutor | Karka LMS

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Linear Algebra

· Examples

✏️ Worked Examples — Step by Step

Easy

Are (1, 2) and (2, 4) linearly independent?

💡 Is one a scalar multiple of the other?

👁 Show Solution

(2, 4) = 2 × (1, 2). They are linearly dependent.

✅ Answer: No (dependent)

✅No

Medium

For A = [[2, 0], [0, 3]], find an eigenvector with λ = 3.

💡 A diagonal matrix has standard basis as eigenvectors.

👁 Show Solution

A(0, 1) = (0, 3) = 3 × (0, 1). So (0, 1) is an eigenvector with λ = 3.

✅ Answer: (0, 1)

✅(0, 1)

Challenging

Trace of A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]:

💡 Trace = sum of diagonal entries.

👁 Show Solution

1 + 5 + 9 = 15.

✅ Answer: 15

✅15

Examples