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## Homework Statement

Suppose that ## \mathbb {V}_1^{n_1} ## and ## \mathbb {V}_2^{n_2} ## are two subspaces such that any element of ## \mathbb {V}_1^{n_1} ## is orthogonal to any element of ## \mathbb {V}_2^{n_2} ## . Show that dimensionality of ## \mathbb {V}_1^{n_1} + \mathbb {V}_2^{n_2} ## is ## n_1 +n_2##.

## Homework Equations

## The Attempt at a Solution

Any element ## V_{1+2} ## of ## \mathbb {V}_1^{n_1} + \mathbb {V}_2^{n_2} ## can be expressed as a vector sum of ## V_1 ## and ## V_2 ## where ## V_1 \in \mathbb {V_1 ^{n_1}} ## and ## V_2 \in \mathbb { V_2 ^{n_2}} ##.

## V_{1+2} = V_1 +V_2 ##

Since ## V_1 ## and ## V_2 ## are orthogonal to each other, these are linearly independent.

## V_1 = \sum_{i=1}^{n_1} v_i \alpha_i ##, where ## \alpha_i ## are basis vectors of ## \mathbb {V _1^{n_1} }## and ## \alpha_i ## are the corresponding coefficients.

## V_2= \sum_{i=1}^{n _2}w_i \beta_i ##, where ## \beta_i ## are basis vectors of ## \mathbb {V _2^{n_2} }## and ## \beta_i ## are the corresponding coefficients.

Then,

## V_1 + V_2= \sum_{i=1}^{n_1} v_i \alpha_i + \sum_{i=1}^{n _2}w_i \beta_i ##, where ## \alpha_i ## and ## \beta_i ## are basis vectors of ## \mathbb {V}_1^{n_1} + \mathbb {V}_2^{n_2} ##.

Thus, the basis of ## \mathbb {V}_1^{n_1} + \mathbb {V}_2^{n_2} ## consists of ## n_1 +n_2 ## linearly independent vectors. Hence, dimensionality of ## \mathbb {V}_1^{n_1} + \mathbb {V}_2^{n_2} ## is ## n_1 +n_2 ##.

Is this correct?